Panic in Year Zero!

Panic in Year Zero, sometimes known as End of the World, is a 1962 science fiction film directed by and starring Ray Milland. The original music score was composed by Les Baxter. It was written by John Morton and Jay Simms. The film was released by American International Pictures as a double feature with Tales of Terror.

The family initially attempts to return to rescue Anns mother, who lives near Los Angeles, but soon abandons these plans as panicked refugees climb over one another to escape the fallout from multiple nuclear explosions. Witnessing the threads of society being torn apart, Harry decides the family must find refuge at their secluded vacation spot.Along the way, they stop to buy supplies, or, in the case of hardware store owner Ed Johnson Richard Garland, take them by force when he wont accept a check. They also encounter three threatening young hoodlums, Carl Richard Bakalyan, Mickey Rex Holman, and Andy Neil Nephew, on the road, but manage to drive them off. ........

Source: Wikipedia