Why would you need to track your oncologist?
I've never had to track him or had a problem of where to ring if I need anything. And before you ask, yes, I'm on the NHS. My treatment is based in Leicester. They have a dedicated cancer centre, like all regions, of which my oncologist is part of a huge team that look after an increasing amount of cancer patients. If I have a problem I ring a 24 hour manned hotline. They then advise me what to do. Usually go to Leicester. If I have an admin problem, and it's quite hard to have one of them seeing as they text and send mail out on a regular basis, then there's a number to ring. It's not my oncologist.
The times I do have to go in to see someone I occasionally get my oncologist. Mostly I get one of his team. They all sing from the same song sheet so it's never a problem. If I really need something then they provided his secretary's number. Just leave a message and they are in contact within 24 hours.
He rings me once a month before my home treatment, to discuss my recent bloods and to make sure I'm ok to have my next immunotherapy. He did this yesterday and yes, I'm still kicking the cancers arse. It's a personable chat and lasts around 5 minutes. There's no rush on his part. He listens to my concerns. If I bring anything up then it's put down for the next blood tests. That I arrange at my local hospital once a month.
Today I've had a call about the delivery of tomorrow's drugs. A nurse will text me in the morning to confirm delivery and time she'll be popping in to administer.
It's all very slick and well organised and for the most part doesn't involve him at all. At least what's visible to me.
As for taking up a place...as you said, most, if not all specialists work NHS and Private. They have spaces to fill. No one will lose out because of this as both systems work independently and alongside each other. I've been to private hospitals for scans. I've been to private hospitals for operations. I have some next week.
If you pay to have private then you will not jump any queue. You will just see less patients around. It's more of a one on one scenario. Your appointment will be on time. Generally. If it's a private hospital it will appear brighter, cleaner and more efficient.
The King though, different proposition. He will have his own surgeon. A surgeon who is probably not doing other operations but spends his time doing lectures and teaching. He probably has a title added, like OBE or Sir. It's another level of treatment in a very exclusive private hospital. One that the likes of you and I, private or not, will never see.