Just like that. No new patients. Tough.
It's only relatively recently that you'll find yourself driving past one with "Now accepting NHS patients" on big banners outside.
I had a lot of NHS work done last year. I didn't pay much because I was used as a guinea pig by a couple of freshly qualified dentists who wanted to improve their skills before being promoted to private practice.
One dentist was good, one was kind of okay, I guess.
The deal wasn't explained. It was "And this is the NHS dentist you'll be seeing." Which is maybe not as ethical as it could be.
I've paid for private care in the past and it's never been significantly better. You get more choice of filling/crown material, but the actual work seems to be as good - or bad - as NHS work.
By far the best work was done for free by supervised students at an NHS teaching hospital in the 80s. Most of it has lasted for decades.
The issue with dentistry is possibly that there's no distinction between primary health care - the "have this tooth out or die of sepsis" kind - and purely cosmetic surgery, which is strictly optional.
There's also some overlap with products like implants, which improve looks but can also work better than fillings/crowns.
I can't think of any other branch of medicine which has the same tension between health and appearance.
I guess parent is a bit more upper-middle class than you.
For this I get 4 cleans/checkups a year and a zealous hygiene regime. I also get some insurance for accidental trauma too.
Sure, if you do need major work it's going to be expensive but the emphasis is very much aimed at preventing this. In the past 15 years or so I have had minimal work done.
Relying on an NHS practice is, as indicated in other replies, a bit of a lottery and you'll be hard pressed to get decent treatment for something serious. Partly because some dentists are not qualified to do stuff like root canal and partly because of funding restrictions - they just don't want that kind of work.
Having said that, things have improved since my childhood when NHS dentists were payed per filling. You can imagine how that worked out in many case.