Having lived and worked close to London for many years, I can say you absolutely suspect wrong. London has several of the largest and highest-quality teaching hospitals in Britain, which means in the world. They are literally world famous. You could maybe google "Great Ormond Street". And they're all absolutely free at the point of delivery.
Here's a thought. If you are rich, in London, and considering having private medical or surgical treatment in a swanky private hospital, do you know how you find out who the best surgeon is, to make sure you get the best? You ask what his NHS job is. You make sure you get someone who is a senior consultant at one of the best NHS hospitals, then you go to his Harley Street private consulting rooms and see him there. He'll attend to you wearing a Savile Row suit, and book you into the Cromwell or one of the other private hospitals, and you can have your hip replacement or whatever in luxurious style, and he'll charge you a lot of money.
Then he'll change out of the Savile Row suit and go to his day job at Charing Cross or St. Thomas's or somewhere, and do the same operation on Joe Bloggs the Heathrow baggage handler. Who will pay not a penny.
And in the event of something going horribly wrong with your swanky private surgery in the swanky private clinic, an NHS ambulance will be called to rush you to the NHS teaching hospital for emergency and intensive care. Because that's where the expertise and facilities are. And if you're a British resident, you won't pay a penny for that either.
Rolfe.