How did the various countries that have a somewhat to entirely public healthcare system transition from what was previously private healthcare?
I know many of the posters here have a specialty in specific countries and regions, so hopefully leaving this question open ended will allow for a diverse collection of answers based on the different areas.
What is interesting here is that the very notion of state-sponsored healthcare, as a component of the liberal states in the 19th century, was in part driven by a desire to strengthen the state against rival structures. It can be said that the nationalization of healthcare was a long process that developed concurrently with the expansion of the modern state.
In pre-unification Italy especially, welfare, healthcare, and education had been in many places either devolved to private associations or handed off to the catholic church. Prof. Vera Negri-Zamagni has edited a volume narrating the transformation of healthcare in Italy which contains a useful essay by Stuart Wolof titled “The “Transformation” of Charity in Italy, 18th-19th Centuries.” (Bologna: Il Mulino, 2000) in addition to series of other papers examining the state's gradual appropriation of welfare.
The earliest healthcare systems in Europe were insurance funds run by professional associations collecting artisans in similar trades, the direct descendants of the welfare systems managed by the old medieval guilds. In Italy, some professional categories, like Doctors and Lawyers, still get their insurance this way. However, it would be during the Second World War and its increased industrial employment that made it necessary and desirable to oblige employers to run similar kinds of insurance funds for their employees, even though because the war took a turn for the worse, the plan was only implemented in 1947. In 1977, the management of these funds was handed over to the INPS, the newly-founded national welfare authority. The pre-World War II ideation of near-universal healthcare is actually ahead of the trend for the rest of Europe, with France and the United Kingdom only adopting a universal system after the war, influenced by the "Beveridge Report," an influential investigation by the British Health Ministry.