I also recommend:

"On the Beach" - still a powerful film all these years later.

"Testament", 1980 or so, when Reagan began his nuke sabre rattling. Could be the best post-nukewar movie ever.

NOTE - "Fail Safe" was remade with TV stars - DON'T RUIN THE EXPERIENCE WITH THIS TURKEY! The film is by Sidney Lumet and stars Henry Fonda as the President. The remake is horrible and pointless.

"The Bedford Incident" is a good nuke-showdown-at-sea movie. There was another starring Gene Hackman and Denzel Washington, forgot the name.

Of all these, "Testament" is the most chilling, "Fail Safe" the most suspenseful, "On the Beach" the saddest and most illustrative of the pointlessness of it all.