If the private hospital is in the NHS England directory to provide NHS care, then if you see a Consultant privately he must put you on the NHS list. If not in the NHS England directory, it is down to the Consultant's discretion whether or not if you see them privately whether or not they will do an NHS referral.
This is very misleading and wrong. What you mean to say is that some private hospitals (BMI, BUPA), also offer NHS services. Your GP can refer you to that hospital as an NHS patient. Wait to see a consultant there might be quicker than the local hospital but will not be as quick as if you were referred privately. The consultant treating you there (who is likely to be the same consultant working at the local NHS hospital) will have to adhere to the same procedural rules, than if treated at your local hospital.
It's NOT jumping the queue. If you see a Consultant privately, then ask to be put on his NHS list, you will wait just as long as a patient who originally went through their GP for an NHS appointment
It is jumping the queue if you see a private consultant within a week, when you would have waited 8 weeks to see the same consultant at your local hospital.
A private consultant suggesting further treatment that the patient can afford privately and therefore requires NHS treatment should refer the patient back to their GP for an onward referral to the local hospital which will mean waiting for the 8 weeks anyway.
Private consultants used to be able to add their private patients to their own surgery lists, but it is becoming rarer as hospitals monitor this practice as it is against the NHS constitution. Consultants who still do so can get into trouble with their NHS employer(s) and rightly so.Why should people who can afford a private outpatient appointment get to be treated quicker than those who can't?
Either you pay for the whole thing or you wait like everyone else who chose or has no choice but to be treated under the NHS.