Short answer: no.

The first hurdle you will encounter is decay of the radioactive elements used in the warhead. For a modern thermonuclear device, the main component to worry about is tritium, which has a half-life of only 12 years. Nuclear warheads need their tritium replaced periodically in order to remain viable. However, you can use a more primitive fission warhead - something using uranium-235 (half-life: 700 million years) or plutonium-239 (half-life: 24,000 years) will still be intact. This isn't as massive a detonation as you're envisioning but it's still nothing to scoff at.

However, the second problem is one of triggering materials. There are two main ways to trigger a nuclear detonation. One is to have two subcritical masses and ram them together really quickly using a (chemical) explosion. The other is to have one subcritical mass and compress it using the shockwave of an explosion. You'll note the key shared word there: explosions. You need chemical explosives to be able to produce the prompt-critical chain reaction to cause a proper detonation.

However, chemical explosives are not shelf-stable over very long periods of time. Even totally isolated from the outside environment, they very slowly decay into more thermodynamically stable (read: non-explosive) forms. Other answers suggest that for conventional explosives such as regular ammunition, you're looking at a period of decades rather than centuries before they're useless. You could salvage the radioactive elements from such a bomb to make a new bomb, but you couldn't detonate it as-is.

Third, as @AlexP points out in comments, there are safeguards built into the design of nuclear warheads to prevent these sorts of scenarios. For instance, there may be an altitude sensor that prevents the warhead from arming until it passes above a certain altitude, or an acceleration sensor that prevents it from arming unless subjected to extreme acceleration - both of these are designed to keep it from going off unless it's attached to a missile and properly launched. You could, of course, override these sensors with care and ingenuity.

Yes, if...

Yes, provided it's not a modern device, for the various reasons outlined in other answers, plus that these devices are designed to fail-safe – if anything isn't working perfectly, you'll not get supercriticality. Even if all the firing circuits were still working perfectly, and he had codes that were still valid, there's absolutely no chance that the conventional explosives used to trigger them would perform as expected after that long.

Key to this is that modern devices don't have enough fissile material to create a critical mass – they rely on an explosive lens imploding a sub-critical mass to a higher density at which it becomes supercritical. That requires powerful, precise and specialist conventional explosives, including very precise timing (computer-controlled, I believe). This is partly for cost – fissile material is expensive – and partly as it's a failsafe – it makes it extremely difficult to detonate it not only accidentally, but even deliberately if you can't activate the firing mechanisms.

But...

But, if the device was a much simpler one, then yes, it could still work.

A gun-type bomb such as the 'Little Boy' used at Hiroshima is very basic. Two sub-critical masses are pushed together to create a supercritical mass.

U-235 has a half-life of 703,800,000 years; Pu-239 24,110 years. So even after 600 years, a little boy style device would still hold enough to create a critical mass.

Obviously the original explosives would have deteriorated, but as these devices are much simpler, you could replace them with whatever explosives are available – the "Little boy" used silk bags of cordite as a propellant, loaded during the flight for safety, so the propellant clearly does not need to be anywhere near the quality or precision used for implosion type devices.

You likely won't get anywhere like the full yield (and remember that the efficiency of these devices is low anyway), but you'd still get a fission explosion. Depending on the original intended yield, that could still be plenty.

So yes, a functional nuclear device could last that long, though not the style used by major players today.

But...

But there's no 'little-boy' style devices around today that we know of, though it's possible that North Korea or other smaller nuclear powers are using such devices.

So you'll need to add something to your history, to posit a scenario where prior to the fall of the nation who created the nuclear weapons, they switched back to simple devices.

• Perhaps this is due to cyber-warfare – they decided that any computer involved in detonating a nuclear weapon was a risk, so reverted to much simpler devices which can be triggered manually or by a simple chemical or clockwork fuse.


• Perhaps the factories or other infrastructure which was needed to produce some of the components were lost, forcing them back to simpler devices


• Perhaps a general/etc. was a bit paranoid of either of those scenarios, and got some simple devices manufactured 'as a precaution'


• Perhaps the device was captured from NK or another such nation, and brought back to be investigated and/or dismantled.

None of those seem particularly unreasonable options if there were a protracted conflict resulting in a nuclear war, the setup to your book.

However, it seems less likely that this type of device would be used on a (high-technology) missile. Potentially the missiles were reverted to older tech too, or perhaps only the bomb-making infrastructure was damaged, not the missiles. More likely, the devices were not installed on a missile, but were intended for use as a parachute bomb (Hiroshima), or a mine; they were just stored in an easily defensible silo for security.

Sort of...

All modern nuclear weapons consist of a primary and a secondary. The primary is triggered using chemical explosives (compression) and an 'initiator' (for an initial burst of neutrons) and has a yield in the low kiloton range.

The energy from the primary is then used to both compress the secondary much more efficiently, and provide the neutrons to initiate the reaction. This secondary can then provide a yield deep into the megaton range.

It is possible to add further stages, and in some (realized) designs, this was done, but for modern weapons this is unnecessary.

Materials

• Plutonium-239: Half-life of about 25 kiloyears. Will alpha-decay into Uranium-235, which, in the amounts that it will appear in, will substitute just fine. You mainly have to worry about the effects of the radiation on the rest of the device, but that's probably manageable.


• Uranium-235: Half-life of about 700 megayears. Has a whole decay-chain with (much) shorter half-lives than itself, but you'll mainly accumulate Protactinium-231 (half-life about 32 kiloyears). Thanks to the longer half-life, it's far less active than the plutonium, so if you can handle that, you'll be fine.


• Deuterium: Stable.


• Tritium: Half-life of about 12 years, decays to helium-3 which is stable, and absorbs neutrons, hindering the reaction instead of helping it. Thoroughly gone by your time, and you can't make it. We can't use it. (on real weapons where it is used, it is kept in canisters that have to be replaced regularly)


• Lithium-deuteride: Stable. Physically that is. You do need to keep it sealed, because it won't like water or oxygen.


• Uranium-238: Half-life of about 4 gigayears. Don't worry about it.


• Polonium-210: Half-life of 210 days. We definitely can't use it.


• Beryllium: Stable


• Chemical explosives: These are one of our big issues. It's unlikely that any of our current high-explosives will remain in usable condition over 600 years.


• Electronics: Even more so. While solid-state electronics are remarkably tough, other components (such as large capacitors) are not. It might be possible to construct a "firing-system-on-a-chip", but I have my doubts that it will last 600 years.

Design

Secondary

The secondary is the easy bit. No chemical reactions are required, so it can be made (and usually is) chemically stable

• Lithium-deuteride fusion fuel


• Plutonium-239 'sparkplug' in the center of the fusion fuel


• Tamper encasing the fusion fuel. Usually made of Uranium-238. Can also be made from Uranium-235 for somewhat higher (and dirtier) yields, or lead, for cleaner weapons (far less fission, so less possible fallout).


• Possibly some kind of 'window' controlling the flow and timing of neutrons from the primary to the sparkplug

The secondary is compressed by ablation of the tamper. The ablation is controlled by the geometry of the primary, the secondary and the casing. A fission reaction is initiated in the sparkplug by a flood of neutrons from the primary's reaction. This fission reaction then provides the neutrons to breed tritium from the lithium and the heat required to start deuterium-tritium fusion. The fusion reaction then provides more heat and fast neutrons to fission not just the sparkplug but also the tamper.

If built with enough margin, there is no reason the secondary couldn't last 600 years and still be functional.

The casing

The purpose of the casing is to internally reflect the radiation from the primary in such a way that the secondary is efficiently compressed. Some devices for controlling the flow of neutrons from the primary to the secondary may also be present.
There is probably some kind of foam filling most of the casing to keep the primary and secondary in position. This is something that will probably need to be replaced. Metal or ceramic struts could also be used for holding things in place, and while that changes the way the radiation flows through the casing, it can probably be designed around, and will last far longer.

The primary

Here the real problems start. There are two types of fission devices: gun-type and implosion-type. The idea is that you want a sub-critical assembly to transition to super-criticality very quickly (well within the expected time for spontaneous neutron emission in your fuel. If the reaction starts too early, it may force the critical assembly apart, ending the reaction and drastically reducing the yeield). The high rate of spontaneous fission (causing neutron emission) from other plutonium isotypes is why gun-type weapons exclusively use uranium.

Gun-type

The gun-type is far simpler, simply assembling two sub-critical parts at high speed using a gun. The downsides are safety (there are a lot of ways it can go off accidentally, or turn into an uncontrolled nuclear reactor), low efficiency (lots of fuel needed for relatively low yield) and a single large dimension (all US designs were over a meter long, due to the gunbarrel).
The minimum barrel length depends on the quality of your explosives and the acceptable risk of predetonation. If you're willing to accept a very long device (several meters) and a higher risk of predetonation, even black powder could work.

Implosion-type

The implosion-type requires highly coordinated detonation of specifically shaped explosive lenses. This compresses the (hollow or solid) sphere of fissile fuel to supercriticality, and greatly increases the reaction rate. Such a design can be boosted (fusion fuel injected into the core to produce more neutrons, increasing efficiency), but this is not required.
What is required is at least two (spherical types use far more, but with clever engineering, this can be reduced) very accurate detonators, carefully machined explosive lenses, and simultaneous ignition of those detonators. Replacing the lenses, detonators and electronics is probably beyond the capabilities of your cult.

Initiator

The reaction in the primary is started by an initial burst of neutrons. In early weapons, this was done by crushing a capsule of beryllium and polonium-210. Later designs used a neutron generator that fired tritium ions at a deuterium target, producing a tiny fusion reaction that generates neutrons.

We can use neither polonium nor tritium. A deuterium-deuterium reaction is feasible, though far less efficient, but whether a fusor device will survive long enough (including driving electronics) is a big question.

A gun-type weapon does not actually require an initiator though. After the parts have been assembled and have achieved supercriticality, the reaction will start, it will just be less efficient if it starts with a few spontaneous emissions, rather than a coordinated flood.

Conclusion

A black powder-driven fission device is probably feasible, if you don't plan on carrying it anywhere. It could have a breach where the Cult leader can place the charge of holy powder prior to firing.

Once you have a fission reaction of sufficient size, it could be possible to use that to drive a secondary. This would require having the barrel stick into the (very sturdy, and very well-sealed) radiation casing. There may be all sorts of reasons why this won't work, but it could be possible.

It may therefore be possible to design and build a two-stage device with a black powder-driven gun-type primary, which can be maintained and fired with very basic explosives knowledge. To my knowledge, no such device has ever even been designed though, and the only purpose I can imagine would be an apocalypse-proof operate-in-person self-destruct-your-entire-city device.

This type of device would have to be designed this way though, you can't just cobble it together from leftover parts without a very deep understanding of how everything works and the ability to calculate what would happen.

If this is indeed what you need for your story, you now get to figure out why on earth someone (or rather, some organization) designed and built a device like this.

Another problem with this scenario is that in an all out Nuclear War, keeping a missile siloed is not a viable option. Silo based missiles are one part of a "Nuclear Trifecta". A country achieves the Trifecta when they have nuclear arms that are deployable by ground based platforms (silo missiles, mobile missile sites (USSR/Russia)) air platforms (Gravity Bombs dropped from air planes... often the first possible delivery vehicle any country can use... certainly the most reliable by your time frame) and submarine based (the last of the trifecta, usually and only reliable second strike method). There are also two phases of an all out Nuclear War: First-Strike and Second-Strike. First Strike denotes the first en-mass nuclear strike and all missiles are launched immediately. It's pretty much the salvo that turns a conventional war nuclear. The Second Strike is the retaliation from the attack of the first strike and is normally launched as soon as a first strike is detected.

The goal of the First Strike is to eliminate a Second Strike before a Second Strike could be launched. Thus, targeting the ground based weapons (silos) was critical to any First Strike planning. It's estimated that no matter which side launches first, the first strike will 3% of all Nuclear missiles capable of Second Strike retaliations on the enemy side. Most of these would be Ground Silo and Air based missiles (mobile Launch Platforms might survive, but Global Thermonuclear Warfair is played very much like Horseshoes: Close enough counts). Subs are stealthy enough and mobile enough (and can move over a larger area) that they are only Second Strike vessels and will survive a First Strike Launch.

For this reason, a Silo based nuclear missile will either be launched immediately or fails to launch and is destroyed by an inbound missile. Additionally, given the secracy of the Subs, siloes are more visible in nuclear deterrence propaganda. Second Strike goals are not so much to obliterate the other side as they are to deter the other side from making a First Strike, as the destruction is mutually assured. It's M.A.D. but hey, we're here and not dying in a nuclear waste land. It's assumed they would get spotted pretty quickly by intel, so they aren't the best hidden sites and for good reason. The U.S. and the USSR both believed the other side was far out performing them in production of delivery vessels. The so-called Bomber Gap and later the Missile Gap, were both issues that featured heavily in Truman, Eisenhower and Kennedy's campaigning. Russia had a more concerning but similar issues, as they knew the U.S. claim to the loser in these gaps were through their own deception and they were really behind and needed to close the gap.

Suffice to say, once the bombs start flying, survival of any bomb is highly unlikely. It's either going to hit something or get hit before it can leave it's silo.