The Fallout series originally started as a result of disputes over who had the rights to the name, with some of the original team eventually working to create a different post-apocalyptic series.
Where did you get that info? The only rights dispute that I'm aware of regarding the original Fallout was in regards to the GURPS system. Originally Interplay had secured the rights to do a CRPG using the Generic Universal Roleplaying System from Steve Jackson games, but they changed it to be their own custom system using the S.P.E.C.I.A.L. traits.
I have never heard that there was a rights dispute over the name "Fallout". In fact, I just watched Tim Cain's video about how the setting of Fallout was developed and he mentioned that the name didn't even get picked until the game was mostly finished.
Maybe you're referring to the later rights disputes that happened in the 2000's after Fallout 2 when there was a discussed MMO and canceled Fallout 3? That's not when the series "originally started out" though.
In that same video Tim Cain said that Wasteland may have been an influence, but the game was more influenced by post apocalypse fiction from the 50's to 80's like
A Boy and His Dog and
Mad Max.