This one makes absolutely no sense when you actually think about it, but the unsettling paranoia of it keeps the show going - to a point. If, for example, so many people in the government know about the impending disaster, why is it that only the small group of families is making an effort to escape? Why does their pursuer not try to leave? Why, if Fritz Weaver's character is worried about being spied on when he talks with his colleague in his basement, does he lay out the whole plan to his wife with no regard for surveillance?
The big reveal of the ending - which is designed to be a "Hmm...that really makes you think" moment is telegraphed ahead. Even if it wasn't, it's a hokey ending and a way out of addressing the paranoia and breakdown of society that accompanies nuclear war. Serling addressed this issue (and in much better fashion) with stories like "The Shelter," "Monsters are Due on Maple Street" and others. Why he shies away from it here is anybody's guess.
Weaver, a great actor, does a good job as the leader of this bunch, but the rest of the episode is forgettable.