You certainly pay for it in France, the national security system will pay 70% and the remaining 30% is for you to pay if you don't have additional insurance. And spend a year out of the country without paying tax and you are not covered anymore. I'm no expert regarding the UK, but I know it is less generous than France.
Lots of people misunderstand the way "nanny States" work, yes they will treat you first and not let you die like in Thailand, but ultimately you are footing the bill.