As regards GP prescriptions...
It can happen. It should happen -- assuming the patient meets the criteria -- but unfortunately it isn't happening often enough. I had to push for it, which I did by writing numerous letters highlighting the relevant passages in the Scottish protocol, the Royal College of Psychiatrists' guidelines and the WPATH Standards of Care. Even then, the initial result of the practice meeting was a refusal. They're cautious, and I can understand why up to a point, but once it's clear that the patient is stable and settled in their new gender role, has no comorbid/contraindicating conditions and has full capacity to give informed consent, it's just frustrating that most are still so reluctant to offer treatment. It's driving people to self med... or worse in some cases. I can understand the need to be certain of the diagnosis, but in clear-cut cases where all criteria have been met, I think general practice needs to catch up.
Ultimately, my SO wrote a letter to the practice manager outlining the debilitating effects dysphoria and lack of treatment were having on me, and this, along with our convincing my GP to contact the GIC and talk to an expert, made the difference. The GIC doctor advised my GP that treatment ought to be an option in my case (assuming that the psych reports were in order, which they were), and I got my prescription almost five weeks ago.
So it's worth a shot, but it's definitely pot luck and will depend a lot on your specific circumstances and how your GP and practice feel about it all. If you think you have a case, though, it's worth pushing for it.