The Trident missiles don't use GPS since they spend their time underwater a lot and GPS signals don't penetrate seawater very well. In flight after launch they use an internal inertial guidance system and in space at the top of their ballistic trajectory they use a star tracker system for final course changes before their descent to glory.
GPS was developed from a predecessor called TRANSIT which was developed during the Cold War specifically to provide US ballistic missile submarines with an accurate position prior to launching their missiles.1 GPS provides this functionality too, as well being extended to other branches of the military, and more recently to us proles too.
Inertial guidance doesn't do diddly squat unless you know exactly where you started from. Whilst Trident missiles might not use GPS, the Trident launch platforms certainly do.
1See for example page 3 of https://www.fs.fed.us/t-d/pubs/pdfpubs/pdf99712826/pdf99712826dpi72pt01.pdf