I will outline the whole process for you:
NHS:
1) See GP
->Couple weeks
2) Referred to local psych to make sure you are not a mental.
->Couple weeks
3) Assuming not mental, get referred to one of the gender clinics, in my case, Charring Cross, London.
->6 months
4)Appointment at clinic.
5) Get asked a bunch of stuff and told the requirements for HRT. Legal name change, RLE experience of 1 year.
->6 months
6)Next appointment
->And so on. Until you satisfy their requirements.
The private doctor I went to:
->Pay ~£250
1) Phone up get an appointment within 2 weeks.
2)Have interview type thing where he assesses you a bit. Agree to have another appointment in 3 months and have a blood test (my GP was willing to get the blood test done on NHS) in the mean time.
-> Pay ~£100
3) Have a follow up appointment take blood tests results. Assuming all normal and you have seen a psych who has confirmed your not crazy within the last 6 months you get hormones.
Pay ~£100
4) Buy hormones for ~3 months (my GP was not willing to prescribe on his behalf so no NHS prescription).
5) Agree to follow up appointments with blood tests every 3 months for the first year at least.
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Hope that helps. I was with the NHS for a while I think I had 3 appointments, even went to my last appointment with them fully dressed and all even though I am not really sure how well I pass. Through London crazy busy rush hour on train. Apparently that doesn't show dedication though because my name is not legally changed and no RLE for a year. So no treatment for me.
Private I had hormones within 6 months and no problems or hassle or unfair requirements. If you have the money I would recommend private every time.