Former Air Force nuke guy here - you’re conflating the process of arming and firing a nuke, and giving the order to do so. The process is strong and designed to prevent accidents and ‘rogue agents’ including psychological or religous problems; who can give the order is the President so arguably we have someone who is devout giving the order.
There is a basic rule for the handling and control of Nuclear Weapons called ‘no lone zone’. This means that you can never be alone with a nuclear weapon, and any arming or firing system for a nuclear weapon requires multiple people to complete the full cycle. For instance, in a single-seat F-16 fighter, the pilot can drop the weapon, but cannot arm it for a nuclear blast without inputting a code that he does not have; he has to get that code from an outside source. In the management and loading of nuclear weapons, we never had less than a 3-man weapons crew, and therefore were never alone with the weapon.
In the missile silos, there were 2 weapons firing officers, who have to work in conjunction to arm and fire the missile. Neither of them can do it alone - there are keys on opposite sides of a large room that must be turned simultaneously in order for the weapon to arm and fire. This is above and beyond the code inputting and such required to even open the arming key safes, which again the crews must get from an external source. In B-52s, the pilot and copilot, or any 2 flight crew officers, must work together to arm and drop any nuclear weapons.
Submarines were the one exception to this ‘external code required’ to arm the nukes. Nuclear missile submarines were ordered to keep a low profile and surface at discrete times and look for a particular signal. If they didn’t hear that signal in X months, they were to fire their weapons at pre-determined targets. This was to be the ultimate deterrent - even if command and control was completely disabled, even if the US was completely taken over or destroyed by a sneak attack, the ‘undetectable’ missile subs dotted all over the oceans would surface months later and fire their missiles and be effectively unstoppable and destroy the Russian homeland. But even then, the weapon system is designed that it requires 2 people to arm the weapons. So even if the captain went rogue, he would still need to have at least 1 accomplice because it would be physically impossible for him to arm the nuclear missiles without that.
As to the question of the OP, yes we were screened for religious views as part of psychological testing for what was called the Nuclear Surety Program. We were also analysed for our basic phsychological fitness and our financial responsibility, as well as other things -they were looking for instability or an avenue for enemy agents or blackmailers to get to us. If I was truly devout and wanted to get near a weapon, I probably could have lied and done it. But the system is designed to make an intentional firing and arming of the weapon by a lone person very very difficult if not impossible.
However, giving the order, as I mentioned above, is down to the civilian leadership and last time I checked there is no psychological testing required to become President and the President can give the order. I’d like to think the system of checks and balances, as well as simple human common sense, means the guys in the trenches would ignore the order if they knew nothing was going on in the wider world, but that’s far from certain.
