The patient would go to the back of the NHS queue, so I'm not sure how they're jumping anything.
The real problem is that private healthcare over-treats people with unnecessary surgery, and cherrypicks the easy stuff, but dumps patients back into NHS care as soon as they get complex.
This is not true.
I went to the NHS over my kid's glue-ear. We had to see a consultant, but the earliest appointment was 9 months, and the hearing problem was impeding our child's language acquisition. So we went private, and had an appointment with the consultant in a few weeks. The consultant recommended surgery to install grommets (wow!), and a few weeks later we attended for NHS surgery, to be performed by the same consultant.
This was 30 years ago; but I've been told recently (a couple of days ago) that this still goes on.