Reckless Rat

I'd say go for it.
Even if you only get an initial look by a specialist, you can still go NHS for treatment based on information you've 'bought'.
I suffered for 15 years with pain and restricted movement in a dodgy shoulder, my GP said I was getting old, that's what happens!
I was only 26!!
My company offered private medical insurance which I took up.

Once I went private, they took X-rays at the first meeting. Even I could see what the problem was, before the specialist saw the X-ray. An operation to carry out sub-acromial decompression on the shoulder fixed the problem. After 15 years of problems and pain I was fixed within a couple of weeks, the feeling was fantastic, no pain and complete movement.

My partner also had a shoulder problem, and went through all the physio, and injections on the NHS which had no effect.
She didn't have the insurance, so she bought the diagnosis privately, and had the treatment on NHS.
It may be different now but the NHS seems to like keep trying multiple cheap fixes that don't always work, when for me, the X-ray (although expensive) would have provided the answer straight away.