Since this is a good scenario for Truman, I'll put in a few more details:
- Richard Russell still runs for the Presidency, and loses badly. Realizing he will never be President, he still redirects his goal to electing Lyndon Johnson as POTUS.
- Truman takes Ernest McFarland as his running mate, which means that Lyndon Johnson is Majority Leader who wants very much to be in good grace with Truman so he can get the nomination in 1956.
First of all, Truman will be a legendary figure, with three amazing election upsets (of course, no one will remember 1940). And the US will pretty much become a standard social democracy as a result of a third Truman term with solid Democratic majorities. His liberal legacy will probably be second to only FDR, though LBJ or Humphrey might beat him if the century goes well.
What Truman will be able to get depends pretty much on how much Johnson can give him without Richard Russell and the Southerners (the source of Johnson's power) deserting Johnson. Truman will probably hint at Johnson getting the top of the ticket in 1956. Johnson was a master at being a conservative to conservatives and a liberal to liberals, so we'll probably see some token conservative legislation that is overshadowed by more prominent liberal legislation. For example, an Atomic TVA could be traded away for Universal Medicare. This is a bad example though, as Truman was quite fond of an Atomic TVA and it would probably happen no matter what.
But with solid majorities, the Fair Deal happens. Medicare
and an Atomic TVA are going to happen. Johnson can do that as long as he remains a stalwart against Civil Rights. But when
Brown v. Board happens (by a smaller majority, whoever Truman puts as Chief Justice will likely not be as good as Warren was there), Truman will put his full support behind it, unlike Eisenhower. Johnson will be in a very precarious position there. The Civil Rights Act of 1954 will be stronger than 1957, but certainly weaker than 1964. One dimension that even Southern Senators were a bit ashamed (relatively of course) of is keeping the right to vote from blacks. Blacks will probably get the right to vote in a weaker Voting Rights Act, and over the next decade Jim Crow will be torn down either by one big law like in OTL, or Southern politicians gradually tear it down due to the new black vote.
Taft-Hartley will be harder to repeal, especially for Johnson. I'm not sure if it can happen, even Johnson failed to get cloture IOTL.
Truman remarked that had he ended Korea like Ike did, then the Republicans would have tried to impeach him. Since Stalin will still die on schedule (most likely), Korea will probably still end on schedule. This will give a huge boost to the McCarthyists. McCarthyism will be stronger in the Republican Party for longer. McCarthy fall against a President of his own party like IOTL, so the Republicans will keep fanning the McCarthy flames. He might still be around by 1956, which is a scary thought, as he might be a dark horse for the nomination, but I don't think it will come to that.
As far as voting coalitions go, the development of the Republicans and Democrats really depends on how the Republicans react to a Democratic President going gung-ho for
Brown. The response will probably be muddy. It really depends on who is nominated for President in 1956, especially because they'll probably win. With the last Republican win in the South being in 1928, they probably won't go for a Southern Strategy, though. Ike winning Texas is what gave the Republicans greedy eyes for the segregationist vote. The three most likely nominees, Warren, Nixon, and Stassen, are all for civil rights anyways. Even McCarthy favored civil rights.
The world avoids a lot of nasty decisions Ike made in foreign affairs. Truman didn't support the whole bipartisan coup business against democratic leaders in the third world, whether it be by the Republican Dulles Bros. or the Democratic Kennedy Bros. Murder Inc. No Operation Ajax against Iran, which is a godsend for U.S. Middle East policy. And Guatemala doesn't undergo decades of civil war and genocide because Truman doesn't approve of the coup there, either. Truman regarded the Republicans opposing Ho Chi Minh even after Dien Bien Phu as fools who wanted to get the U.S. into a war in Indochina. Without Eisenhower, a united Vietnam becomes the Asian Yugoslavia. Truman will make an anti-Mao friend rather than divide and (fail to) conquer Vietnam. With Vietnam as our friend, there's obviously no Vietnam War, which is HUGE, but it also could stop Sino-U.S. rapprochement. Truman won't be a fan of that anyways, having his presidency nearly ruined by a war with China. Truman will probably come down even harder against UK-France-Israel in Suez than Ike did.
A lot of modern Presidential precedents go back to Ike and won't exist. Billy "Friend of All Presidents" Graham, and the National Prayer Breakfast are gone. Camp David remains Shangri-La. The Presidential stereotype of golfing often is gone.