Some YouTube links from Metafilter and casual Googling:

The War Game (1965): [Part 1] [Part 2] [Part 3] [Part 4] [Part 5]

When the Wind Blows (1986): [Part 1] [Part 2] [Part 3] [Part 4] [Part 5] [Part 6] [Part 7] [Part 8]

Miracle Mile (1988) [Part 1] [Part 2] [Part 3] [Part 4] [Part 5] [Part 6] [Part 7] [Part 8] [Part 9] [Part 10]

The Survivors: Start here.

And if you really want to be depressed:

Threads (full movie). Ten times bleaker than The Day After.