As a general principle, all space-based hardware is "hardened" against radiation, at least in the sense that someone will have run the math on the probability of a radiation event causing undesired behaviors such as single event upsets. Informally, everything that goes into space is @&#$ expensive, and is used in situations where people really don't want anything to go wrong. It warrants taking the extra time to check for such occurrences.
As for military hardware that remains on the ground, as a general rule it tends to be more robust to extreme environments than normal equipment. For instance, MIL SPEC integrated circuits often has extreme temperature ranges specified which normal corporate customers find no need for. They cost a pretty penny because of this!
So the real question is not whether an event could cause a mishaps for these systems. The real question is, if an event of sufficient magnitude occurred to disrupt these systems... what else went wrong?
I would expect an event which permanently disables military hardware, with their extreme requirements, would have far reaching implications for the States that might be firing the nuclear weapons in the first place. Trying to win a nuclear war after such civil devistation would be like trying to finish winning a correspondence game of chess immediately after being knifed in the back or having a large boulder fall on you. It isn't a game nations like to play.
Which leaves the question of whether a nuclear device could go off without intent. The general answer is probably the most resounding "no" you can get short of when the LHC announces that they are 99.9999% positive they have discovered something. These are not unstable devices. Nuclear fuses are designed to be markedly reliable to a level of pathological fervor that most of us humans don't fully comprehend. Consider, as an example, the first nuclear bomb dropped, and its safeguards.
- The bomb was not fully built at takeoff. The cordite charge that would drive the projectile rings of uranium towards the target rings (causing the nuclear explosion) was not even installed until after the plane was already in the air. (This appears to be the choice of the weaponeer, not necessarily a design choice)
- The bomb had a physical cutoff preventing power from going to the charge's detonators unless an arming plug was installed. These plug were about 1 inch in diameter and dumb as can be, by design. The armed plug had wires bridging the gaps between the pins to complete the circuit. The safe plug (installed to keep the socket clear of debris and to make sure nothing could accidentally touch the pins and bridge the circuit) simply had no wires. Even then, there wasn't one plug but three. All three arm plugs had to be installed for the bomb to work. This was done on approach to the target by climbing into the bomb bay and physically swapping them out.
- There was a 15 second timer, which ensured no fusing event could possibly occur until 15 seconds after drop (defined when the electrical plugs connecting the bomb to the plane were physically puled by the bomb dropping). This ensured that any false positives which could occur in the later fusing steps could not possibly set the bomb off. Again, dumb as possible, by design. (no solar event was going to stop either of these first two steps, short of the sun expanding to swallow the earth 4 billion years from now).
- There were two redundant radar altimiters measuring the altitude of the bomb. These had the actual control over the firing switch which actually sent the charge to the cordite.
- These radar altimiters weren't even trusted. There was a barometric fuse added as well, which measured air pressure and would not let the altimiters fire the bomb until it was satisfied.
And this was all features applied during the heat of a world war. Modern weapons have safety features constructed over decades. Modern nuclear weapons do not go off unless someone wants them to.
Which brings me to the one caveat to the whole "it can't happen" argument. Some nations, such as the UK, have a "fail deadly" nuclear arsenal. Unlike the US, which strictly limits the authority to fire its estimated 4,018 nuclear weapons to the president and only the president (considered "fail safe"), the UK took a different approach. They have a smaller arsenal (estimated to be 120-215 weapons, depending on which stat you look at), primarily in the form of nuclear tipped Trident missiles to be fired from their four Vanguard submarines. They, as a policy, keep at least one at sea at all times. Unlike the fail safe arsenal of the US, which explicitly withholds some of the launch keys until the president gives them out, the fail deadly arsenal of the UK stores those keys on-board the Vanguard submarines when they are on duty. Each captain is given an envelope, the letters of last resort, instructing them what to do with these keys should they lose communications with the British government.
These letters of last resort are handwritten by the prime minister them self, after the prime minister is briefed on precisely what damage a Trident missile can do. They are then sealed in a nested double safe onboard the submarine, and never opened unless needed. At the end of a tour, or if a prime minister retires, they are brought back, unopened, and summarily destroyed such that only the prime minister themselves ever knows the content of the letter.
What is written in these letters, is of course secret, but it has been publicly revealed that there are four known options for a captain of a Vanguard:
- Retaliate with nuclear weapons
- Do not retaliate
- Place your submarine under the command of an ally (the US and Australia have been publicly listed as possibilities)
- Use your own judgement
From this, it is clear that there is a possibility that these letters of last resort could lead to an individual being given the authority to exercise judgement, so thus could indeed intend to launch a nuclear weapon. No nuclear safeguard is designed to stop an authorized individual from consciously launching a weapon, so if an event wrecked nuclear defenses and the letters of last resort were written in a particular way, Vanguard subs could launch (this probably implies that the solar event looked like a nuclear attack). They typically carry 48 weapons, and current estimates suggest 100 firestorms are needed for nuclear winter, so two such subs would both have to be given letters which lead them to fire their entire arsenal.
So the final question would be, which nations have fail-deadly arsenals, and whose defenses might need to be negated to cause a nuclear winter. That is a more complicated policy. While the UK publicly acknowledges its fail deadly arsenal, as a deterrence policy, its not popular to admit that's your approach. Russia has the Dead Hand system, which was designed as a fail deadly system. Most sources state that it is not active, only activated in emergencies, and its exact details are not all public, but it could be a sufficient piece of the puzzle.
After writing all of this, I have to reflect on your 95% depressing nuclear post-apocalypse. If I may quote Douglas Adams:
What to do if you find yourself stuck in a crack in the ground underneath a giant boulder you can't move, with no hope of rescue. Consider how lucky you are that life has been good to you so far. Alternatively, if life hasn't been good to you so far, which given your current circumstances seems more likely, consider how lucky you are that it won't be troubling you much longer.