Yuppers,
Panic in the Year Zero - 1963 Nuclear war aftermath, family of four with Ray Miland and Franky Avalon.
The Quiet Earth - Zac Hobson was the main character, scientist working on "Project Flashlight" - Big oops! (Filmed in New Zealand)
Not saying you're wrong, just making conversation. I wouldn't consider either a Cold War movie.
Not only does Quiet Earth have one of the coolest final shots in movies, it has a marvelous end title music cue. I'd say it's more metaphysical than Cold Warian. If that's a word.