Obviously not happening in real life, but in the film The Iron Giant, a nuclear missile is fired at a small town in Maine. They have about 5 minutes of warning before the bomb comes. As the characters debate what to do, one of them suggests getting to a shelter. Another responds "It wouldn't matter."
My question is, would it matter? Could you be beneath enough concrete to protect you?
The Nautilus didn't carry submarine-launched ballistic missiles. In fact, the United States' first ballistic missile submarine, the George Washington, didn't enter service until 1960.
In the movie's timeframe, the US Navy had only the Regulus Cruise Missile submarines Tunny and Barbero, the latter assigned to the Atlantic in 1957. The Tunny was in the Pacific, under the policy of having one Regulus submarine in each ocean. Purpose-built Regulus submarines weren't deployed until 1958.
The Polaris A-1, carried aboard the George Washington, had a CEP of 5,900 feet and a 600 kiloton nuclear warhead. The CEP is important. That means only about half of the time, the missile would land within a mile of the target. Now, /u/restricteddata had a visualization of CEP, but I can't find it at the moment. So let's refer to his NUKEMAP simulation.
I've selected Bangor, Maine as an example. As you can see, even landing at the edge of the CEP ring, more than a mile away from the target, the town would be well within the 20 psi air blast radius. That's enough to severely damage or destroy reinforced concrete buildings. It's also well within the radiation radius indicating 50-90 percent death from acute effects.
In addition, most civilian shelters were built as fallout shelters, not blast shelters. They wouldn't do much good in the event of a nearby blast. Blast shelters did exist, but to be within a mile of a Polaris A-1 warhead, you'd have to be in something well-constructed, like Mount Weather or Cheyenne Mountain. I don't think it's likely that there'd be something like that in Maine.