Panic in Year Zero! (1962)

While on a fishing trip, Harry Baldwin (Ray Milland) and his family hear an explosion and realize that Los Angeles has been leveled by a nuclear attack. Looters and killers are everywhere. Escaping to the hills with his family, he sets about the business of surviving in a world where, he knows, the old ideals of humanity will be first casualties. Not one to give up, he holds up a store for supplies and hides the family in a cave.