Found the perfect PoD area -- in 1915, the American Association of Labor Legislation drafted a
sample bill, which effectively provided the same kind of health, sickness, and death insurance found in Germany, and submitted it to various state legislatures. (Interesting note: the AMA initially supported these measures.) Though there were efforts in something like eight states (including the four largest of the time: NY, PA, IL, and OH), California probably came closest; however, they ended having their referendum on the question of establishing UHC at just about the worst possible time for any piece of progressive legislation, in November of 1918. If any of the major forces working against the law were removed -- say, the anti-German scare that OTL accompanied US entry into WWI -- I can easily see at least one of these measures getting passed by 1920, with Cali as the leading candidate.
Also something that might have potential -- maybe if Washington state's
Initiative 172 -- which, FWIG, provided healthcare for those on government aid -- hadn't gotten
derailed by Red Scare politics, budget gridlock, and the 1950 counter-referendum.
Bismarck started state-subsidized care in Germany sometime in the late 1800s as did Sweden, as I recall.
You're right about Bismarck -- his government passed three laws in the 1880's that pretty much established the German healthcare system. Sweden however,
FWIG, didn't really have anything like UHC until after WWII (which is when a lot of other countries we're starting to adopt and experiment with
their UHC systems)...