I still disagree with the notion that where you studied medicine would come into question with private practice. At the end of the day if you have been able to make so far in your career that you are highly qualified and experienced enough to carry out private practice and pass the relevant gollege examinations, then where you studied becomes absolutely irrelevant. I know plenty of plastic surgeons working in the private sector who have trained abroad at medical schools in Pakistan and India and I know two from Africa and they earn loads (I assume). The public has never really perceived Medicine at Oxbridge as being superiour. I think when going to a Doctor for treatment the last thing on a patient's mind is where the doctor studied.
And also I think it would reassuring for patient to know they are receiving treatment from an individual who has had years of experience in their field and has carried out many successful operations in the past whether on the NHS and/or on the private sector. This to me as patient would be far more important information than where the doctor obtained his/her medical degree.
I think the people who are saying that it does matter really need to scratch beneath the surface of this topic. You cannot deny the fact that there are non-Oxbridge private doctors and those from medical schools you wouldn't consider as traditional. But where does the comparison end? What are we considering as traditional? London and Oxbridge? If so does that mean we don't regard medical schools such as Belfast and Edinburgh as "well-respected" (remember those two are extremely hard to get into)?
I think in being employed for private practice your track record and personal qualities are far more important. You could be an Oxford graduate but have failed examinations after university and may not have had the same level of experience as say the BSMS graduate. I think the employer (assuming he/she is a medic) has more sense than to simply discriminate on the basis of university.
If it was a case of going to a particular university made all the world of difference every medical applicant would apply to Oxbridge and London. That just isn't the case.