Is it possible for a proper labo(u)r movement to arise in the US (i.e. one that isn't just shot at by the police and then abandoned, but actually gets into the political establishment) in the early 20th century? Jello Biafra's TL seems to indicate that this is possible, but how likely is it on a scale?
Depends on the POD. Keep in mind, in OTL the U.S had a rather substantial labor movement that was growing quite quickly at the turn of the century - about 4% of the non-ag labor force in 1900, 8% in 1910, and 16% in 1920 (keep in mind that the UK labor movement was at about 10% prior to WWI)- and was beginning to move past its initial political isolationism due to the need to legalize labor unions, leading to a tacit alliance with Woodrow Wilson's Democratic Party.
If the anti-union drive known as the "American System," which coincided with the end of progressive rule in the Federal Government and the arrival of the Red Scare, hadn't happened (or if a more protracted American involvement in WWI had required the Federal government to buy the loyalty of the labor movement with more permanent protections), then the U.S labor movement wouldn't have lost 1 million members (with the AFL taking 20 years to get back to its former level of 4 million members). With more of a secure footing, the CIO and the AFL could well have expanded beyond the 30-odd percent of the workforce clustered in the industrial Northeast and Midwest, which would have greatly boosted both their economic and political power in the crucial decades of the 1950s-1970s.
Alternatively, there's an earlier POD that could have made a huge difference to the development of the American labor movement. Unlike the British labor movement, which was legalized in 1871-4, American unions were effectively made illegal through the Sherman Anti-Trust Act and subsequent Supreme Court decisions, and the courts were used systematically between the 1870s and the 1930s to smash unions through injunctions. This in turn fostered an attitude of voluntarism and reluctance to organize politically, both in the AFL and the IWW. If instead, the right to organize, strike, and boycott had been recognized in the 1870s-1880s and American unions had been widely seen as legitimate and legal organizations, the labor movement would likely have grown much faster in the late 19th century - the Knights of Labor stood at ~1 million members in the 1880s, but American labor was down to half that in 1900.