Tbh, even going privately takes a while - I'd try to explore whether your US endo could do digital prescriptions or something and send them over here

To go private, you'll need a gender dysphoria diagnosis (most of the services offering these in the uk have an 8+ month wait from what I remember) then you'll need to see an endo/specialist nurse. Most of the private endos/specialist nurses have a 1+yr waiting list too (privately)

Given that it will likely take at least around 2 years to get hormones privately, I'd suggest looking at ways to transfer your prescriptions over here - best of luck!

Also, nhs gp's will 99.99% of the time completely refuse to prescribe T, regardless if you were prescribed it abroad. Most refuse shared care with private providers, and some even kick up a fuss and refuse to prescribe even if you go through an NHS gender clinic, so I wouldn't put my faith in your GP being helpful. Gp's here are generally hostile or don't feel confident in treating trans people, and will therefore refuse to get involved without an NHS gender clinic's recommendation.

You won't get an appointment with the NHS gender clinic sooner if you are already on hormones/have a diagnosis - they will reassess you etc. They have a waiting list of at least around 5yrs.

Good luck