There were a number of interesting Cold War movies made, and a lot of drek. I recently highlighted The Day the Earth Stood Still. That is probably one of the best 1950s-era Science Fiction movies, (This is era that gave us Plan 9 from Outer Space, after all) and one of the better Cold War movies.
Seven Days In May is a political thriller based around the possibility of a military coup taking place in the USA. It isn’t perfect, but it has a good cast, and Rod Serling (of The Twilight Zone) wrote the screenplay, and he knew how to keep you on the edge of your seat. Hmmm. A story about removing a sitting President because you don’t like him or his politics. Why did that come to mind?
And the whole film is a reminder that the phrase “It can’t happen here,” is just not true. So here’s the trailer from the 1964 movie. They apparently remade this (sort of) as a TV Movie in 1994 with Forest Whitaker.