No experience with gynaecology but this is my experience of going private:
Most consultants work both NHS and private. You actually want to see one who does both as a consultant who does private only is likely to be weird.
Going private - you get to choose who you see. Or at least we did as we are self- funding. Spend a lot of time researching who you want. So for example I wasn't getting seen at all for my severe migraine - going private, I saw the head of the migraine clinic at the National Hospital for Neurology with in a week. I was never ever going to get to see him on the NHS.
You get seen much much quicker although they do of course have holidays, they are still human beings!
They have very helpful and accessible secretaries.
If you need scans/surgery etc the wait is minimal. DH waited 18 months for supposedly urgent surgery on the NHS. Privately it was 3 weeks and they apologised for a wait which was absolutely beyond their control.
You get seen in weird clinics where there is always a coffee machine, and an array of posh magazines - Vogue, Tatler, Conde Nast Traveller. Instructions in the loos may be in Arabic.
You pay for everything. Depending on what you are going for, it can get very expensive, fast. However everyone we have ever seen has acknowledged we are self-funding and not taken the piss. A lot of piss-taking goes on if you are on insurance with unnecessary investigations, referrals and followups to boost up the money - I am a doctor and numerous consultants over the years have told me how the game is played.
Sitting in a waiting room once we saw an elderly lady explain how her lovely consultant still wanted to see her every 6 months to check her hip was OK - even non-medical DH whispered 'I bet he does, kerching!'
The ideal self-funding condition is one with minimal tests, and minimal follow up that once it's fixed you are done. A lifelong condition is spendy. However the bonus compared to the NHS is you see the same consultant each time, your appointment is never cancelled or postponed, you always get the right tests - you may think this is worth the spend.
Finally you aren't limited to what can be done in that clinic. They work in networks so you might see the consultant in one clinic, have a test somewhere else, and operation somewhere else and so on.