Otto von Bismarck, in his position as Reichskanzler (he was the 1st Chancellor of Germany), created the first form of a modern welfare state with a universal healthcare insurance (1883), accidents insurance (1884) and pension system (1889) in the six years between 1883 and 1889. His plan was to combine socialist ideas with religious values and corporate interests.
Specifically regarding universal health care: Bismarck decided to implement a two-tier system of national healthcare ("sickness") insurance and private healtcare insurance, where everybody had to be insured at least in the national insurance. Civil servants and people above a certain income threshold were insured in the private insurance.
First rudimentary approaches to this universal law were policies in Prussia dating back to 1845.
Bismarck's 1883 law didn't include rural workers, so I'm not sure to what degree it can be considered universal. Certainly significant, but not quite that wide of a net. It certainly created the foundation for a broader scope of social insurance, so if we're looking for a good historical starting point for the idea of state-sponsored social insurance programs, 1883 is it.
And while Bismarck's immediate motivation for pursuing Sozialpolitik largely stemmed from his desire to mute the appeal of the SPD, we should recall Christopher Clark's advice that, "to personalize the issue is to miss the point." The idea of national insurance (at least in Germany) grew not out of Bismarck's Realpolitik, but from the young historical school and thinkers like Gustav von Schmoller and Adolph Wagner. The Long Depression (1873) had demonstrated the need for a more hands-on model for the state, and these ideas resonated on both the right and the left.
Grimmer-Solem, Erik. The Rise of Historical Economics and Social Reform in Germany, 1864-1894. (Oxford, 2003)
As far as I can tell, the first country to introduce a universal means-tested system was Denmark in 1891, but my expertise ends at Schleswig-Holstein, so this is based solely on secondary materials.
Culter, David and Richard Johnson. "The birth and growth of the social insurance state: Explaining old age and medical insurance across countries," Public Choice 120(2004).
Maynard, A. Health care in the European Community. Pittsburgh (1975)