Well, when we did it, it was 1946. We were (economically as well as literally) rebuilding the entire country from the ground up at that point anyway, having expended not just the entire country's, but the entire Empire's wealth and resources on fighting the war. So there was essentially no entrenched rich lobbyist class to protest against the welfare state; Labour, promising to implement the Beveridge Report's recommendations on setting up a welfare state, had been swept in with a huge majority (ousting Churchill from Number 10!), and the Health Minister most associated with setting up the NHS was one of 10 children and left school at 13 to work in a colliery. The medical profession fought him every step of the way; he claimed (says Wikipedia) to have eventually won them over when he "stuffed their mouths with gold". This presumably works less well on mouths already stuffed with gold.

How you arrange for similar circumstances in the US I don't know. The Great Depression might have been a good opportunity, but this current depression, painful as it is among everyone up to and including the merely comfortably-off, is unlikely to wipe the slate of the very rich clean like World War 2 did here. And in the US, with a few Old East Coast exceptions such as the Kennedys, political power has always inhered in wealth rather than in birth as in the UK.

One other thing that did occur to me: you (that's "you" the US in general) have voted Democrat majorities into the House and the Senate, Democrats most or all of whom presumably stood on a platform of health-care reform. You then voted a Democrat president into the White House, who very definitely stood on a platform of health-care reform. Yet despite a clear majority of you voting for it twice, meaningful health-care reform is still not looking likely. Something is not working properly in your democracy.

Peter