The Day After was an American made-for-TV movie, from 1983 which starred Jason Robards, JoBeth Williams and a very young Steve Guttenberg (pre-Police Academy). It had a very simple premise: what happens during a nuclear war.
While it wasn't as gut-wrenchingly graphic as Threads (which was made the following year), it did have some chilling moments - university students see the missiles launching near campus, and realise that no, this isn't a drill, and that most of them have maybe half an hour to live. A child flash-blinded when he looks at a fireball. A father explaining to his daughter that they can't take the family dog into their basement shelter, because they won't have enough food for it (the dog is later seen lying dead outside the basement door).
When I first saw this, at age 17, it made me sad rather than scared, and I have no problem watching it now. It lacks the stark horror of Threads, but it was quite radical for its time and context - these were the early Reagan years, remember - and did try to show some of the consequences of "pushing the button".