The Overlord wrote:
But would an Axis victory have prevented the Earth from becoming involved in Intergalactic society? Would the people of Earth never have overthrown the tyranny of the Axis powers? Would defeat in WWII for the Allies have meant that Earth will be dominated by the Axis powers for centuries to come or is there something else that Axis victory would have done to prevent Earth from joining the Federation?
Also, there's no reason to assume the Nazis would have remained a presence in the 2060s or the 2160s. It would suffice that their brief histrionics in the mid-20th century changed something about the Earth that then either stayed changed, or then caused a cascade of events.
I believe the Blish novelization of COTEOF suggested the Earth tore itself apart in rebellion after the Nazis won. No more Axis, no more Allies, no space flight of any kind.
How Spock could have derived that using stone knives and bearskins is another question.