The consensus in London once was that the doctors who couldn't hack it in the NHS went to Harley Street.
That hasn't been my experience, I would say it is more or less exactly opposite. When I first came over, i had US insurance and was able to go private for care (w/ little cost to mysefl). Generally, the private/Harley st doctors I saw were top class. In the NHS, i have largely seen newer doctors (either younger or immigrants) or less competent and older doctors.
Afaict, when you are starting out, you go w/ the NHS to repay schooling and build a client base. If you are any good, you eventually decide to make real money and start a private practice. So you have the chance of getting a really good younger doctor but the odds are against you. Now, our obstetrician was fantastic and she did work w/ the NHS once per fortnight. (I was never sure if that was out of altruism or an effort to maintain some sort of standing in the NHS). So there is a chance of getting a really good private doc when you go through NHS but the odds are a bit longer on that.
Now this was generally docs for pregnancy/infant care, minor illness and a few minor hospital visits. Maybe intensive/ER care is different, I (thankfully) cannot speak to that.
Don't get me wrong, i think the NHS is actually a great system but the private docs I have seen have been much better than the public ones.