America pays more, but they can also generally see a specialist for any field within a matter of days or sometimes hours for non-emergency care.
It only costs a few extra quid per month to get a Bupa (private company) top up to smooth out all the areas where the NHS is a bit rough. The end result is that we still end up paying less than you do in the US since the Bupa cover does not have to worry about the really expensive stuff as the NHS covers that.
With a Bupa policy I can leave work now and go and talk to the private clinic just down the road with no appointment. Even without a Bupa policy I can still go and talk to them and then just pay cash for their time (I have done this to get some jabs for going to South America on short notice).
In contrast, I sat in a hospital with my daughter, her hand oozing blood after being crushed and having two broken bones and severe lacerations and soft tissue injuries, for nearly THREE HOURS before being treated. Watching her quietly cry the whole time I would have gladly written an American-sized cheque for £1000 to have her treated immediately.
Then why didn't you: http://www.bmihealthcare.co.uk/hospital/description?p_hosp_id=294&p_hospital_page_id=163&in_page=Emergency%20care%20centre
It took me about 30 seconds to find that link via Google. I bet they have other clinics all over the country.
I know A&E is awful on a Friday night through experience, but that was when I was a poor student dropout and I broke my hand (cleanly snapped third and forth metacarpals). If I had been in the US at that time I probably wouldn't have had any cover either as my family are not that well off so god knows what would happened.
As it was though I was in surgery within 16 hours and then discharged within 36 at no cost apart from the taxes I had paid as a checkout boy in a supermarket (the only job I had ever done at that age). Now I happily pay my taxes as I am grateful to be able to type this using both hands and go climbing as a hobby thanks to my 100% recovery because the NHS pinned my hand back together.
In the US I very much doubt I would have been able to get my hand reconstructed the same way without paying a fortune, instead they would have just patched me up so it did not get any worse but left me with limited movement.