NHS psychologists might see some clients privately too. I'm not sure... I know that it was common back home for people to work in the public service and then have a day or two working privately.

Have you asked your doctor for a private referral? It might be that he doesn't know anyone... Can be hard to tell with referrals, sometimes they are little more than 'you scratch my back and I'll scratch yours'.

The upside is that you will get some choice rather than being stuck with who you are assigned to.