When I worked for a gynaecologist, the usual thing was that I'd be processing the referrals to send to Appointments and then he'd pass me another GP referral letter and tell me to book her in on the next available appointment - it would be somebody who had seen him privately but he was now adding onto his NHS list (as they have/had the right to do).
Essentially, if they paid to see him, he'd do the first consultation and then tell them what they could access on the NHS - if it was suitable for their needs/wishes, he'd see them there instead and if not, they'd stay private patients.
The other advantages are shorter wait, convenient appointments that actually happen within 5 hours of the appointment time (or at all), not having to deal with Registrar gatekeeping - which largely seems to be in my experience being told there's nothing wrong with you, you stupid woman, you're clearly mentally ill to think that you've got an autoimmune disease that you've been treated for since you were a child, read up on fibromyalgia and go on a diet - then they deign to look at your notes and say 'oh, you have been treated by a consultant for this before. Would you like a steroid injection now or would you like to see the consultant first?' - and no sitting in knackered seating (if there is any/that isn't occupied by non patients) with ten other clinics all running 3 hours behind in the same area, including multiple screaming children, somebody eating McDonalds and three elderly and at best confused patients dumped in the general area in hospital gowns asking random people for help to go to the toilet.
Gynae also tends to share space with Antenatal in the NHS - which means social deprivation effects upon the general atmosphere (thinks back to days where discreetly rescuing scared older ladies and leading them to my office for safety when there were ex partners, GM-to-be and current partners meeting outside the ultrasound room people expressing displeasure over injunctions, care proceedings and 'It's nothing to do with you, you aren't going on the birth certificate' discussions broke out ).
Not that I could ever afford it, but if I could, private healthcare would be my choice. Just to avoid having to deal with the limitations of the NHS at close range. And the inherent sexism of people who wouldn't be like that if their income depended upon treating you as a competent human being.