Panic in the Year Zero (1962), starring Ray Milland, Jean Hagen, Frankie Avalon

Synopsis of Panic in the Year Zero

In Panic in the Year Zero, Los Angeles is demolished by a nuclear attack.  Looting and robbing run rampant. Seeking shelter from the chaos, a family hides out in an isolated fishing cabin.  Only to find that the chief danger comes from a band of violent, crazed juvenile delinquents.