Congrats!

The wait very much depends on the clinic, and also on your local health authority. The clinic will process your referral in due course, but GD isn't routinely funded on the NHS so they have to apply to your local health authority for exceptional funding before they can see you. Only after your local authority has approved your funding will you get an appointment date from the GIC.
If you've been referred to London, you might be in for a looooooooong wait I'm afraid, not just for the first appointment, but also between subsequent appointments. But many of the other clinics are quicker.
I was originally referred to London but after waiting 7 months without receiving an appointment (they hadn't even processed the paperwork), I cancelled my referral there and was referred to another clinic. I saw them two months later, and after 3 assessments over a period of 18 weeks (6 weeks between assessments) I was cleared for hormones & surgery.
All-in-all, the time between coming out to my doctor and starting hormones was about 14 months. And that took a
lot of pushing.
The good news is that the rules have changed recently (August) so if you want to start HRT you can ask your GP to put you on 'bridging endocrine treatment' while you wait for the clinic to come back to you.