Well , I'm living in UK and I do have the same problem more or less. Usually you are referred by you GP to the nearest clinic of your area (example for my case : Nottingham with a waiting list of 10 month)
The main issue we all believe we have to go to our nearest GC well your GP could enquire to any gender clinic founded by the NHS where the waiting list is less long. A lot of GP's in UK are not aware of that.

If I were in your position I would ask for Northampton.

Well , in my case I decided to go private for my HRT and councillor, until the NHS wake up. I'm extremely patient.

Shorter waiting lists mean shorter time till surgery and free laser,  even if you are doing HRT private.

You right Kira , I will pay a visit to my GP next week for him to enquire and request him for Northampton GC , let see  it cost nothing to try.
For laser most is done now, maybe one session left and after that I will have certainly to for electrolysis (white hair make me feel so old)