Jenny,
There are many GIC's throughout the UK, many which do both NHS and Private appointments. Most PCT without a GIC will refer through Charring Cross.
Both of course have there benefits and drawbacks, the NHS route at times can be long and beuracratic, but of course funded by the state. The Private route is much shorter, but offers a wider choice of surgeons. but obviously can be expensive and if you start this route (and later go into the NHS) it can be very problematic.
I know many people who have sucessfully been through the NHS system and also many who have gone privately and achieved their aim.
I went privately through Leicester and transitioned in just under 14 months, from legal name change to SRS. I had Speach Therapy on the NHS and my GP took my private prescription and I got hormones on the NHS.
I really does depend on your circumstances, your ability to fund therapy / surgery privately and being realistic in what you wish to achieve in what timescale.
I hope your transition goes well.