As previously mentioned, I had my PSMA scan privately at the Paul Strickland Scanning Center which is a separate Charity based facility within the grounds of Mount Vernon Hospital although there is some cooperation between the two and indeed a Consultant at the hospital interpreted the scan for me. Some hospitals are able to 'call the scan over' but a good idea to get it on a DVD so you can make it available to whoever you choose.
The scan cost circa £2400 in May 2018 which was about the going rate at the time, although the Royal Marsden were asking £3K+ then. There are several hospitals in the London area offering the PSMA scan but a few other hospitals now doing it around the UK. I think Andy's suggestion of placing yourself under the care for treatment where they also have PSMA scan provision is a good one. Just remember apart from treatment as and when required, there is further visits for checkups, MRI etc.
I had no problem having my PSMA scan privately but being under the aegis of a NHS hospital where I went on to have more treatment.