‘The 300 Spartans’ (1962), ‘Zulu’ (1964) and ‘The Great Escape’ (1963) are a few of my childhood favourites you might enjoy, based on the films you’ve listed there.

The Ipcress File

The Man Who Would Be King

The War Lord

Panic in Year Zero

Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb
or the more serious counterpart:
Fail Safe

Colossus: The Forbin Project

(Not so old, but belongs here with those other nuclear war films: Threads (1984).)

The Andromeda Strain

Taxi driver

Cool hand luke

North By Northwest

I suggest working your way through the film noir top list.

Laurence of Arabia

Double Indemnity. You’ll never look at Fred McMurray the same way again.

Sunset Boulevard

Philadelphia Story is still one of my favorites.

Chinatown