Much of that depends on who rules Germany after Hitler's death, since Mussolini's rule wouldn't fundamentally change the west's view on racism.
Goering's reign would probably make casual racism more commonplace in the west and would weaken the movement for desegregation in America. Goering's public persona and his overall demeanor would weaken the influence of the numerous KKK groups, and instead inspire southern politicians to take more 'civil' approaches towards pacifying the Civil Rights movement, with plenty of aid from J. Edgar Hoover and the FBI. Southerners might even make state-owned internment camps for 'unruly' African Americans in order to keep them away from the public light and to reestablish order with white supremacy. This will no doubt cause an uproar in the North, but the federal government will likely have little incentive to intervene, for their intervention against well-disciplined Southerners could cause bloody race riots from angry Southerners, or could even lead to a second civil war if it gets ugly enough. Case in point, a Goering win would likely set desegregation by several decades.
Himmler's reign would rightfully be condemned by the vast majority of American politicians, including in the South. But there would be a sizable minority of Americans that would emphasize with Himmler specifically, for his background as a chicken farmer and his headmaster demeanor. A handful of Southern businessmen, as well as a few Klan leaders, would want to replicate Germany's successful 'exclusion' of European Jews not only to cement white supremacy in the South, but to enrich their pockets as well. And while Himmler was certainly no friend of Christianity, he wouldn't waste the opportunity to destabilize America and reduce it's threat to the Reich. But he would have plenty of conditions in exchange for his funding: they had to expand their targets beyond just African Americans (such as Catholics, immigrants and especially the Jews), they had to become a well-disciplined unit of multiple vigilante groups (they could lynch whoever they wanted, but they had to keep property damage to a minimum) and they couldn't provoke or engage with Southern, nor federal, authorities (for that would've killed this plot in its infancy). The result would be a far bloodier Civil Rights era, with these vigilantes killing protesters and causing widespread bloodshed on national television. And Southern governments would be very reluctant to intervene, but the federal government would be increasingly pressured by their western allies to rectify the situation. Himmler would personally order the early assassination of Dr. Martin Luther King Jr., so that America would be engulfed in race riots. These vigilantes would eventually be detained and desegregation would probably still happen around the same time as OTL, but America would suffer considerably more trauma as a result of Himmler's meddling. And he no doubt would use this to legitimize his reign over Germany.
But Goebbels' reign would very likely accelerate desegregation. His venomous and volatile rhetoric would paint Germany as an unapologetic dystopia, and Goebbels would viciously point out America's own racial turmoil if they ever spoke out against him. Even the most racist Americans would've been horrified by Goebbels' openly anti-Semitic beliefs and of the widespread violence that he would've committed against his own people. Few Americans would've been able to stomach enforcing such measures against African Americans, nor would they want to ever be compared to the irredeemably vile regime of the dwarfish firebrand. Southern governments, albeit reluctantly, would've heavily reduced the Jim Crow laws so that enforcement wouldn't be as rigid. And the federal government would've introduced stronger civil rights bills earlier than OTL. But this doesn't mean that the racism and resentment from white Southerners will have gone away. There would still be bitter feelings towards African Americans and even occasionally violent. But desegregation as a whole would've ended early if Goebbels succeeded Hitler.
Very little would've changed had Speer, Hess or Bormann taken over. Just had to mention that.