The Mitchell home is located in a suburb 50 miles from New York City. When Gladys goes to the basement to do her laundry, we see the flash of the New York bomb through the window, and the shock wave from the explosion arrives only eleven seconds later. The shock wave would actually move at approximately the speed of sound, about 700 miles per hour. It should have taken 50/700th of an hour, or almost four and a half minutes, to reach Gladys's house.
The radio says that the weapon used on New York City was carried by a guided missile launched from a submarine. The first ever missile launch from a submarine hadn't happened yet when the movie was made, and sub-launched missiles were ballistic, not guided, for decades.
With the electricity off, the door buzzer wouldn't be working twice.
When Barbara comes home after being exposed to potentially radioactive rain, Dr. Lee, an expert in atomic science, doesn't tell her to discard her clothes, scrub her body, or cut her hair, which are elementary precautions for dealing with exposure to radioactive fallout.