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?
First, Earth didn't join
the Federation, Earth created
the Federation, was the prime factor, was the driving force, in the Federation's "birth."
The movie First Contact says that it wasn't just Humans engaging in warp flight that brought interaction with the Vulcans, the first warp flight had to happen within a pretty narrow time window. Would the Nazis have hit that window? And would the Nazis have permitted the Vulcans to keep Human out of interstellar matters for a century, or would the Vulcan's have been told to "step aside?"
Would the Nazis have even tried to form a early coalition? There a certain amount of conjecture that it was Archer's action in forming peace between the Vulcans and the Andorians that started the Romulans on the path to war, the Romulans didn't want peace between the players. And it was the Romulan War that help the Humans in convincing the Federation's founder to form the Federation at all.
There was nothing about the Federation's formation that was inevitable.