I agree that defining the first UHC is hard. It seems easiest to figure that we'll need to at least get the pre-cursor of UHC passed in the 1910s for it to evolve later.
I'm wondering whether Roosevelt is really well-suited to the role. Yes, the
1912 Progressive Party platform calls for something like health coverage.
"The protection of home life against the hazards of sickness, irregular employment and old age through the adoption of a system of social insurance adapted to American use."
But it also calls for a lot of other stuff that would be pretty radical: an assault on states' rights, popular referenda on a host of issues (including amendments to the federal constitution). Politically, though, I would imagine that TR might get big business to go along with the scheme because it would be a sop to workers. Throw up a big nationwide strike or two, and add in TR riding to the rescue with a government sponsored health system. The difficulty then becomes the constitutionality of the proposal. TR will have to fight for this, as FDR did for the constitutionality of the New Deal. It could well happen, but I'm skeptical.
The problem with all of this is that it's a lot to cram in to 1912-1916, while TR will also be trying to figure out how to deal with the outbreak of a European War (assuming that things still go crazy). Unlike in the 1930s, there isn't really a crisis point in 1912 to rally the public, in part because of TR's legacy from his first two terms. What I do think could happen is the enactment of the following proposals:
Health We favor the union of all the existing agencies of the Federal Government dealing with the public health into a single national health service without discrimination against or for any one set of therapeutic methods, school of medicine, or school of healing with such additional powers as may be necessary to enable it to perform efficiently such duties in the protection of the public from preventable diseases as may be properly undertaken by the Federal authorities, including the executing of existing laws regarding pure food, quarantine and cognate subjects, the promotion of vital statistics and the extension of the registration area of such statistics, and co-operation with the health activities of the various States and cities of the Nation.
An early Federal Department of Health could be quite powerful in supporting some kind of federal health care provision and/or government sponsored/mandated coverage. This kind of proposal will be easiest to pass in the interval between 1912 and the outbreak of a general European war. If you couple greater US involvement in that war with a Spanish Influenza that exerts a bit of terror in the American psyche (which wouldn't be too hard), then you could easily allow the newly created Dept of Health to champion the cause of some kind of basic, mandatory, universal health coverage in the aftermath of the crisis. I would guess that this coverage would focus on mostly matters of public health (routine exams, vaccinations, etc), with specialty care left to a private system of pay-as-you-go. In the 1910s, though, the public coverage may also exclude "catastrophic care" but you could tack that on pretty easily as a later expansion. The crucial point here is using an alternative WWI's proclivity to expand the scope of government, rather than conjuring up willy-nilly a coalition in support of UHC.
However, I'm not sure any of this really serves the OP. The fact that the Progressive are supporting "adoption of a system of social insurance
adapted to American use" suggests that they are specifically hoping the mirror policies adapted by others (like Bismarkian Germany or even the National Insurance system in Britain). The key to satisfying the OP on this score would be require an earlier change.
An essential component to this change is an early challenge to the notion of laissez-faire government. This would remove the cultural/constitutional objections. The best point in time to make a change like this is the Civil War / Reconstruction. There are a few ways to make the change. One is to toy with the pensions given to Civil War veterans, including health care of some form. I'm not sure why they would though. Again the spectre of a pandemic may be useful, though the battlefield medical advances of the war itself may be enough. This won't really change enough, IMO, unless you change something big: the nature, tenure, and caliber of Reconstruction. I personally like the notion of 1) a slightly longer war (meaning a slightly longer draft and some more veterans), 2) Lincoln surviving to serve his full second term (with a bit of court packing) and/or Seward taking over rather than Johnson (that is, someone who won't cede leadership of Reconstruction to Congress, but someone comitted to keeping it going), 3) a Reconstruction radical enough to make something like the Freedman's Bureau stick a bit longer. You could even throw in a third term for Lincoln (though I personally like the notion of him becoming CJSCOTUS). Keep him around for a while, and I think Lincoln might well become something of the proto-socialist that Turtledove depicts in
How Few Remain. (The free labor ideology that led to his support for a "free soil" position (which evolved into abolitionism) might well evolve to attack the depredations of the Gilded Age against workers' rights (as being potentially corrosive of all rights) in a manner similar to how he thought slavery had the potential to hurt the rights of free labor.) A longer draft might mean that the US Gov't comes to a conclusion similar to that in the UK during the Boer War (that many of the masses weren't healthy enough to serve) -- though I've never come across such a reference during OTL's Civil War (perhaps because medicine wasn't quite advanced enough to call for the health guidelines that the UK was trying to meet in the 1890s). An established system of veterans' pensions and support for freedmen means a tradition of government social programs that is continued (rather than abandoned).
Another feature that may or may not be convenient is to keep the Unionist Party around for longer, which may dislocate the political landscape enough to create a space for a more viable Socialist Party and/or Populist Party later in the century.