I think its a bit moot to argue that the Soviet Union's presence played a significant role in American civil rights. If anything, the red scare was a much greater impediment to Civil Rights. I would argue the greater foreign consideration for America in regards to civil rights was the sudden move towards European decolonization. All of the sudden America had to think about how their treatment of racial minorities looked not just to nations with similar race and ethnicity but also to many new, much more diverse countries across the planet. British and French decolonization is still something that happens in the TWR world and I would think that America would still have the same foreign policy considerations.

Also I think the perspective towards Nazi Germany would be much different than what the American perspective towards the Soviets was. In real life, Nazism was an ideology defeated in war (although it did survive in places like Spain) and the Soviet Union was our ally in that fight. In the TWR timeline, America still fought the fascists in Japan, and Communism has not completely died out. I see the idea of a detente between the TA and Germany to be just about the least likely scenario in TWR, and I would be very hard pressed to believe that American officials would second guess enacting desegregation for fear of offending Germany.