Even we children knew that we'd be atomized quickly, should the Cold War turn hot and go nuclear. We would sometimes discuss it and, being so tied to the military, we were pretty acutely aware of those periods where the level of tension changed.
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.