SueDonim I think many people only use privare health facilities for specialist consultations and minor treatments, rather than for in-patient treatment.
As for paying, I do not think many people have health insurance, unless through work. More and more people, even retired people, pay from savings or income. The kind of visits we have paid for, which is mainly consultations may well cost less than £500.
We opened a savings account about 10 years ago that we put a fixed sum of money each month, specifically to pay for such consultations. After 10 years there is a significant sum of money there, that we see as dedicated to paying for any private healthcare we need. However, It is all ours to do what we will with and if need be the money can be used for other purposes if necessary.
The last time we used it was last summer after NHS incompetence brought our daughter to the edge of death. Even after they rescued her at the last moment, her GP was still unwilling to dsicuss the underlying problem with her, what treatment was likely or when it would happen. We paid for her to talk to a consultant. He worked in the local hosiptal so could access her medical records. He discussed what the problem was, the range oof solutions and which would be the besand most likely in her case.
This consultation gave her peace of mind and understanding of her medical condtion and its treatment. She continued her treatment on NHS. She neither jumped any queues, nor got any special advantage from it. I think it cost us £350.
If a doctor was seeing private patients in NHS time, either he wouldn't be in the hospital at his usual clinics, which would draw peoples attention, or he would have to feed them in to the normal NHS system, which would mean other staff being complicit with what he was doing
It is possible a patient paid for a private consultation as we did for DD and then continued with the NHS and waited their turn like everybody else, but when seen by the consultant, would be known to him and some of the necessary tests may already have taken place and the results were available at the appointment.
Unfortunately without Kali2s evidence, we have no means of knowing.