Pretty sh1tty. When a decorated WW2 veteran was lynched in 1946, it was the last straw and renewed efforts were made to outlaw lynching. Shortly after Truman desegregated the military he passed the nation's first anti-lynching laws. Princeton University in 1947 began to regularly admit black students, as a US Navy program during WW2, which had sent a few blacks to PU, proved successful.
In the late 40s the Civil Rights Movement started in its earliest stages. In New Jersey, for example, the new 1947 Constitution outlawed racial discrimination in public institutions- the last of NJ's segregated public schools were disbanded. It isn't very well known, but much of the North in post WW2 was by and large segregated and blacks were banned from many hotels- even black performers in Atlantic City had to stay at black boarding rooms outside AC.
While not a direct correlation with WW2 and the black veterans, the nation was finally waking up to the absurdity and immorality of racism.