NATO's provocation: https://en.wikipedia.org/wiki/Able_Archer_83
Soviet response: https://en.wikipedia.org/wiki/RYAN
We were rather morbid kids and I think our leading child-theory was that they made us do that so that it would be easier to count the bodies.
This would have been in the sixties. I'd later go to a boarding school that was really remote, and they didn't make us do the exercise. In hindsight, getting under the desk while being that far from any likely blast might actually have afforded us some protection from falling debris.
(Also, for nuclear bombs specifically, hiding under the desks keeps kids out of view of windows, and hence not vulnerable to being blinded/burned by a flash.)
But most schools aren't, and they aren't going to roll out those procedures unevenly.
Like the saying goes, "almost doesn't count except horseshoes, hand grenades and nuclear warheads."
Growing up through the Cold War was definitely interesting. It's probably a hell of a thing growing up under the constant reminder that, at any second, we'd be gone and probably wouldn't even know it. I don't know any different though. It certainly puts the small stuff into perspective. Thankfully the Cold War was anticlimactic!
Yet, I don't think it really bothered us. I'm not a psychologist but I know that children still laugh, play, and sing while they are in a war zone. We were not in a war zone, obviously. But, we were pretty normal children and did pretty normal children things. We were just aware that we'd all die if they dropped bombs or missiles on us.
There's probably an actual psychological term for it, but I don't know it. Children are remarkably resilient. Other than the drills, the PSAs, and the silly educational videos they made us watch, we weren't that concerned. We were kids, it's not like we had the power to change anything.
I'm pretty sure there's a psych term for it.