The NHS was not created out of luck, but from the dual factors of the social change which derives from total war and the effective political efforts by the likes of Beveridge, JB Priestley and the Labour Party while the Conservatives were exclusively focused on the war.

The economic argument against the likes of the NHS and full employment were exceptionally strong and near universally accepted and supported pre-war. Post-war they were seemingly entirely disproven. Pre-war very few people would countenance effectively conscripting doctors into working for the state, post war it was far more concievable. Pre-War, the Tories were the proper party of government and responsibility, post war they were the guilty men that failed to rearm while Attlee and Bevan had served effectively and loyally. In short, Total War against the forces of evil tend to help the people advocating more intervention, as happened in the First World War and the effective destruction of the Liberals.

For all their many faults at the time, the Conservatives took the war exceptionally seriously. They effectively suspended all political activity, chose a leader who was not actually a conservative, gave large numbers of influencial positions to Labour politicians, and were exceptionally open to left wing suggestions if they added to the war effort. They also commissioned the Beveridge report and accepted the vast majority of it in vague terms. On the other hand, trade unions and Labour activists were comparatively more involved in politics at the time. Add into this the number of socialist and left wing public individuals given key propaganda roles for the likes of the BBC, and you end up with a much stronger narriative during the war that people are fighting for a better post war world.

The result in 1945 was a Conservative party who had basically not done proper politics for nearly a decade, discredited, but basically accepting the majority of the Labour Manifesto, and a Labour manifesto designed not to actually be implimented but rather to force the Tories to make concessions, because they didn't actually expect to win. When they did, it gave a clear mandate to creating things like the NHS, one the Tories were unwilling to oppose in any meaningful sense until Thatcher, which we call the post war consensus. Furthermore, at this point the Liberal Party was almost entirely dead and buried, and irrelevant to national politics. Liberalism, as we would think about it, was gone.

It wasn't fluffy liberalism, effective argumentation, or statistical evidence that created the NHS. It was the cynical blackmailing of the Conservatives by public intellectuals, a complete lack of faith in politicians, and a weak opposition that did, lead by a Labour party filled with boring aristocrats.

Never mistake for fortune the result of good strategy.