This one was called: “The Shelter”; it was pretty creepy! I looked it up by the words “twilight zone episode dealing with bomb attack” (I think)and found it! A couple of the details in the description in my post were a little bit wrong, but not much. One of them was the attack.
It was the warning that a couple of UFOs were seen flying toward the town. Later, it was announced that they were harmless satellites, and the alarm was called off! I guess it was believed at the time the show was made-early 60s-that those objects could be carrying a nuclear bomb or something.
Oops, sorry for double posts; I got confused, thinking I hadn’t already posted!
This one was called: “The Shelter”;