How much of an impact did Truman's integrating of the military have?
Well, just about everybody agrees that Truman's integration of the military had an affect on the civil rights movement, but that comes three years after WWII. I'd also quibble with the idea that the civil rights movement starts half a generation after WWII, because I think it had peaks and valleys and changed in orientation over time, but I think it would be a mistake to write it off during the war. I would be curious to hear anecdotes about how white troops interacted with their African-American counterparts when working side-by-side. I'd always heard that white officers tended to treat them with a fair amount of scorn in no small part because a substantial part of the officer corps was drawn from the American South; Patton gave African-American units like the 761st a chance by allowing them to fight, but he also to brutally long deployments that other white units simply didn't have to endure, and that drove up casualty rates to a frightening level.
To my mind, and perhaps I will be corrected or enlightened, WWII's significance on the civil rights movement lies in no small part in the number of African-American veterans who come back and begin directly participating in civil rights organizations. Medgar Evers was a WWII veteran and becomes active in Mississippi's NAACP. The leader of Clarksdale's NAACP chapter, Aaron Henry, was also a veteran, and that's discussed at some length in Francoise Hamlin's Crossroads at Clarksdale. Charlie Cobb Jr.'s This Nonviolent Stuff'll Get You Killed directly traces participation from African-American veterans into the success of the civil rights movement, in no small part because many of those veterans had few compunctions about meeting violence with violence. Cobb's broader point about guns in the civil rights movement rubs some people the wrong way, but he does seem to make pretty clear that veterans were emboldened to speak out.
Then of course there's the backlash against African-American veterans. There's a spate of attacks from 1945 onwards against African-Americans, and that definitely plays a mobilizing role in the civil rights movement. Harry Truman had been a member of the Klan in the early '20s and harbored some racist views of his own, but he also tended to idolize veterans. His response in particular to the deliberate blinding of a veteran named Isaac Woodard is nothing short of horror, and those attacks helped to push him to embrace a strong civil rights platform, including the desegregation of the armed forces. While it takes years, well into Eisenhower's administration for desegregation to be accomplished, it signaled both the increasing intolerance of the federal government for overt, systematized racism (in no small part because of Cold War pressures) and the fact that the Northern wing of the Democratic Party was unwilling to tolerate Jim Crow.