Having experienced it, I'm surprised that Brits feel that way. When I was studying abroad a while ago (which was not a great experience for other reasons... https://jakeseliger.com/2012/11/18/about-the-university-of-e...), I needed an x-ray and then a CT scan for what appeared to be a minor issue. So I went to the clinic and signed up for one, only to discover a week later that an x-ray typically takes 6 - 8 weeks to be done! I asked around and was told to go to Bupa facility (https://en.wikipedia.org/wiki/Bupa). To the Brits this seemed like an obvious solution, and paying in cash is in fact a good way to get things done quickly. It turns out that when people have something really wrong with them, the solution is often to just pay out of pocket!
This disabused me of any notion that publicly-provided healthcare is automatically superior.