The 1948 Progressives were too closely linked to the Communist Party to survive in Cold War America, especially after Wallace himself renounced them once they failed to condemn North Korean aggression in 1950. (The one Congressman who supported them who was re-elected in 1948, Vito Marcantonio of East Harlem, lost to a D-R fusion candidate in 1950.) The 1924 Progressives might theoretically have survived--at least as a vehicle for presidential elections--if both major parties had continued to nominate conservative candidates. But (1) it's hard to see who their candidate would be in 1928--"Fighting Bob" La Follette was dead, and his son "Young Bob" was only 33 so not constitutionally eligible. Far more important, (2) organized labor gave up on the party after the 1924 defeat and reverted to the Gompers policy of "rewarding labor's friends and punishing its enemies" within the major parties. (The AFL had never really favored a third *party* anyway, just an independent La Follette candidacy.)
The 1912 Bull Moosers had the best chance, but only if the Democrats nominated a really conservative candidate, which was very unlikely. Either Wilson or Clark--probably even Bryan!--would hold the core Democratic vote, which was enough in a three-way race. The only thing I can think of that could elect TR *as a Progressive* (of course he might win if he got the GOP nomination) would be some terrible last-minute scandal uncovered regarding Wilson or whoever the Democrats nominated. But if TR won because of that, his victory would probably be considered a fluke, and Democrats and Republicans in Congress would see little need to change their party affiliations. (Presumably there would be more than the 13 congressmen elected as Progressives in OTL, but probably not very many more. Observers at the time noted that the Progressive vote had "an 'inverted pyramid aspect.' It is largest at the top and 'tapers down very fast.'"
https://books.google.com/books?id=FJ5FAQAAMAAJ&pg=PA997 This is not *totally* fair--two Progressive candidates for governor, Albert J. Beveridge in Indiana and Oscar Straus in New York, won more votes than TR in their states--but in general the Progressives did lag behind TR in down-ballot races.)