In most cases its the same doctors working in private hospitals and for the NHS.

And in some cases it involves NHS facilities too. There are definitely some questionable aspects of our public/private healthcare. But again, this isn't always how it works, and the ethics involved are also complicated.

I know someone whose condition rapidly deteriorated while receiving treatment in a private hospital and when things were looking really bad they got transferred to an NHS hospital.

I hope the person you know was OK. That transfer is actually not surprising. Emergency/intensive care is almost always NHS here.

Private options tend to be better in situations like having multiple treatments available where one is significantly more effective or more reliable but also costs a lot more. Sometimes the NHS will only offer a cheaper but inferior alternative, which sounds horrible until you think that there is a huge but ultimately pooled budget and any time policy allows more spending on one treatment there is always someone losing out somewhere else.

The US has safety nets as well, they're just not as generous as the UK's.