On the Beach, Stanley Kramer, 1959,
An American submarine sets out to investigate after atomic waste has destroyed most of the world.

Dr Strangelove, Stanley Kubrick, 1963
Crazy American General, Jack D. Ripper, unleashes a nuclear strike force against Russia because he is convinced that they are tampering with his nation's water supplies. Even the sinister Dr. Strangelove is powerless to stop the eventual holocaust

Fail Safe Sidney Lumet, 1964
American planes are sent to deliver a nuclear attack on Moscow, but it's a mistake due to an electrical malfunction. Can all-out war be averted?

The Day After, Nicholas Meyer, 1983
Dramatic portrayal of the horrors of a nuclear holocaust.

On effects of nuclear war:

Testament, Jane Alexander, 1984
Nuclear war in the United States is portrayed in a realistic and believable manner. The story is told through the eyes of a woman who is struggling to take care of her family. The entire movie takes place in a small suburban town outside San Francisco. After the nuclear attack, contact with the outside world is pretty much cut off.

Black Rain (aka Kuroi ame), Shohei Imamura, 1989
The story of the aftermath of the Hiroshima bombing, based on Masuji Ibuse's novel.