I don't remember ever worrying about a nuclear holocaust as a kid, until I saw the Day after and Threads! Originally made for television, The Day After still holds up pretty well today. The depiction of radiation sickness is what really bothered me as a kid. I remember when we would have civil service drills in grade school and I remember taking them a bit more serious after seeing The Day After. I also remember having a lot of nightmares about sirens going off and then seeing a bright mushroom cloud in the distance. I suppose a lot of kids my age had similar nightmares.
I watched Threads a few months later and found it to be much more disturbing. It's a British film depicting what a nuclear exchange would be like and how much the world would change after. It quite a bit more graphic and I remember a disclaimer was run just before the film started.
If you haven't seen either of them it's a good glimpse into some of the paranoia (much of it warranted) of the time.