Testament is a good one. Sort of the same subject matter as "The Day After," but more thoughtful, less sensational, no special effects.

The The Internet Movie Database (IMDb) has a "plot keywords" feature. Bring up any movie (e.g. "Testament") and click "plot keywords" in the left column, and it will bring up a list of words that are pertinent to the plot (e.g. "apocalypse," "radiation sickness," "nuclear war," "disaster"). Then click on any of those words and it will bring up a list of movies with the same theme.

Here is the results list for "disaster."