As far as I can tell, there are two private gender identity clinics in the UK, Transhealth and Gendercare.

Does anyone here have any experience with them, what they are like, how they compare, are they any good etc?

I went to Transhealth last Aug - October. Even though I had to wait 2 months for an initial consultation and a month for an appointment to see the gender therapist (due to an influx of new patients I was told) it was still obviously an improvement over the waiting times for the NHS. The therapist I saw who was female was very nice, informative and very easy to talk too. I was nervous at first but she was good at putting me at ease. Dr Curtis was exactly as described to me. He's straightforward and tells you what's what when it comes to possible regret due to medically transitioning and the effects of T.
It was also cheaper that I thought it would be. The first and final consultation was I think £140 and the therapy appointments were £75 each.
All in all I'd recommend Transhealth. They were very professional, have years of experience and much less rigid than NHS GIC.