Most of the private specialists in the UK work in the NHS as the "day job".
For instance my OS usually works in 2 local NHS hospitals for the day job, but he does private work at a private hospital which is where i had mine done.
I chose to go private as 1) i have private medical insurance and 2) i didnt want to wait as i had private medical insurance.
Essentially because i went private i got to have the head NHS man - the consultant do the operation, whereas in the NHS you would have one of his team doing the operation.
The beauty of the UK is that so long as you can wait, eventually the NHS will get round to sorting you out - it may not be perfect in terms of having to wait for your operation, but generally the care and skill of the surgeons/nurses/doctors etc is second to none. I've had 2 operations on my right knee since Sept, the last operation was to repair my ACL.