You don't need GPS to aim nuclear missiles. They typically use a combination of inertial guidance and star sighting. So we can still nuke people after Brexit, even if the US says no! Of course, if relations got that bad the USN would be shadowing our missile boats from the minute they left Faslane and could likely just sink them.
From the Wikipedia page on the Trident navigation system:
"It is an Inertial Guidance System with an additional Star-Sighting system (this combination is known as astro-inertial guidance), which is used to correct small position and velocity errors that result from launch condition uncertainties due to errors in the submarine navigation system and errors that may have accumulated in the guidance system during the flight due to imperfect instrument calibration. GPS has been used on some test flights but is assumed not to be available for a real mission. "