First of all, the requirement to see a local mental health specialist before going to the gender clinic has been removed, the GP needs to directly refer you to a gender clinic, the gender clinics deal with the psychiatrist assessment directly now. For GP's it is important to educate them on the current NHS protocol, as many dont know what the current thing is, and some dont even know that the NHS provides gender services, i was lucky to get someone who has helped people with gender issues before.
For Charring Cross Gender clinic the waiting time before your seen is currently around one year, for that reason i am getting privately assessed now, and getting the medication on shared care, so i only pay private rates for assessment. The reason why the waiting times are so long is that gender care services receive nowhere near enough funding due to people not really understanding the issue. Someone i spoke to before in south wales was referred to Charring Cross.
At the moment you do not receive any confirmation letter or anything until you have made it through the waiting list, although it sounds like efforts are being made to change this.
You are expected to see atleast two professionals before they prescribe you medication, so probably a Psychiatrist, and a Endocrinologist, and will only be happy with giving you HRT if evidence of some social transition has taken place, such as for me, the psychiatrist was happy that ive grown my hair out, and have neutral clothing, and therefore have a androgynous appearance. He was also pleased with the fact i have given names some thought and settled on one, he doesnt expect me to get it legally changed just yet however. The focus is on having a plan for gradual social transition as the HRT does its work. They wont be happy with putting you on HRT if the plan is an open ended wait and see what happens approach. Note however that they work on a case by case basis and are there to help you, so its in your interest to be open and honest with them and let them do their work.
Note that the person i saw is a consultant gender psychiatrist at Charring Cross Gender clinic, and does occasional private clinics on weekends, under the name GenderCare. They process people exactly how they would on the NHS. You can self refer to GenderCare and by extension i assume most other private clinics like it, although it helps to notify your doctor of the process so they are expecting the letter from the Gender Psychiatrist when the process begins to request shared care.