At least in Germany, you can opt out for private health insurance. I don't know if that's the case in the UK. There're also many private hospitals in Denmark. And in the latter, you have right to get treated in those if there're no public healthcare options available in reasonable time. But both Germany and Denmark suffer from the same issues as the UK, of course.
That is to say, private care is often available in the West. It just comes with a hefty price tag.