It's very difficult isn't it.
I moved house recently and was struggling to get registered with an NHS GP practice. While the health board was basically forcing me onto a surgery I needed to see a doctor, so I sucked it up and went private.
It's expensive. I paid £160 for a 15 minute appointment, and this was one of the cheaper ones -- the other well-regarded private GP in my city charges £250 per appointment.
The other thing to bear in mind with a private GP is that anything you are prescribed you have to pay the full price for. My medication was a bog-standard antibiotic so wasn't expensive, but some medicines are extremely pricey.
Also, in my experience it's even more disjointed than the NHS GP system. The private GP I saw took no history, and couldn't see my NHS history (obviously, cos he wasn't NHS). The result of my visit won't log on my NHS health record either, because it was done privately. So the system isn't joined up.
I went in convinced that the private GP was the way to go, and now I'm unsure. It was quick I was able to see someone the same day privately. But it was also very expensive and I think if I had more complicated needs I'd be concerned about medical continuity. But the NHS provision here isn't great either the usual issues; can't get through on the phone, they've shut down the online booking, you get seven minutes if you're lucky, etc etc.
So I'm torn, but on balance sticking with NHS for the time being.