I paid £400 (maybe more, I can't remember) for an appointment at the Priory a few years ago. Doubtless it would cost more now. I left with a prescription for a beta blocker. Pointless.
I used to have an excellent psychiatrist though but he is sadly now retired. It may be worth looking to see if any of the universities are running trials and get in touch directly with the lead researcher (this is what I did, years ago, and ended up with excellent help for my mental health condition with said psychiatrist). Maybe look on the Royal College of Psychiatry website and in the medical schools for Imperial, UCL, Kings and even further afield.
Also, to be fair, I have just moved areas in the UK and am under a decent mental health team here. I'm basically flabbergasted at the difference to where I was living before (where I was told, "Well, we can't do anything as there are no psychiatrists in the entire county." !!!!!!) The psychiatrist was excellent (ie actually listened to me and didn't patronise or seem impatient with me - which is the more usual experience if you are lucky enough - from my experience - to get to see an NHS psychiatrist).
Good luck. Mental ill health is a difficult path to have to tread.