These are some classic films produced during the Cold War as public service announcements concerning Civil Defense. Though you can laugh at them now, at the time, this was considered truly matters of life and death.

Radiation is as dangerous to your livestock as it is to you. Also, it is not nearly as frightening as these puppets.

The absolute classic: "Duck and Cover" with Bert the Turtle.

From 1969, "In Times of Emergency," hosted by John E. Davis the one-time National Director of Civil Defense.

If you want more, here's a nice list of "chilling" nuclear war films. I, myself, remember watching Special Bulletin and The Day After when they aired....does that date me a bit?