Every time I read a thread like this, I'm so glad I'm not in the U.K. any more. The NHS is failing it's people but it's not the fault of its staff but of a government that refuses to fund it well and public who are resistant to paying in more.
A US system is not the only alternative - there are plenty of examples in Europe that are better than the UK's - but we pay more for them. There is the Australian model, which seems like a bit of Europe and a bit of US. But yes there would be issues with rich/poor areas - where I live in France is basically a medical desert because we are on the Swiss border and HCPs earn more in Switzerland, so they work there.
The French system is really good in many areas. 70% is paid for though taxation and you can chose to pay the other 30% yourself as you go or get a top-up insurance to cover the difference. But à GP visit costs €25, so not expensive. Many medications are pennies. The cost of insurance isn't high.
The Swiss system is great, but expensive, though poorer people get help to ensure they have basic health insurance. I access it through international insurance and it's worth every penny. My DH has special cross-border insurance. Between us we pay about £10,000 a year - I'm not sure how much he would pay in the U.K.. I have had 7 surgeries there, all booked on a day of my closing, mostly within a month of the consultation (a couple within a week!). Follow-up care is amazing. Now under a wonderful pain clinic where the staff are like a second family.
DH worked in Sweden for a year and again they pay more in taxes (taxation is high anyway) but healthcare is almost free at the point of delivery - for outpatient treatment certainly everything is free after the first €100 each year.
Missed appointments have to be paid for in many countries if you cancel with less than 24 (sometimes 48) hrs notice. Things like falling ill with Covid would be an exception. It's an incentive not to waste your HCP's time.
If the U.K. switched to private, the same staff would be providing the care in à hopefully more efficient manner - but undoubtedly extra staff and hospitals would be needed too - there is no quick fix.
OP i hope you get seen soon.