Actually, no reason McCarthy couldn't be the McCarthy of a Kalterkrieg timeline, assuming he was as unprincipled and opportunistic as people say. He just changes targets.
What would be interesting is what the Black Scare targets. In OTL, there really were Hollywood writers and directors who were under Party discipline (ie, taking orders from Moscow) even into the 50s. I don't believe there was any great Nazi penetration of Hollywood at any time. OTOH, expect the Ku Klux Klan to be targetted as sympathizers.
The resistance to the Black Scare will be different than what it was to OTL Red Scare. In OTL, J. Edgar Hoover was able to convince some of the ACLU for a while that Communists didn't deserve their protection. It'd be even easier for the ACLU to rationalize not protecting Nazis--because, you know, Nazis. On the other hand, overzealous persecution of the Klan could mean whole Southern state governments begin opposing HUAC.
Now--what would the 60s counterculture be like? Young people would probably still react against Cold War conformity and paranoia--but Cold War conformity to different ideals, and paranoia about different enemies.