The original history before the events of First Contact, Cochrane made the first warp test flight by himself without the assistance of the Enterprise crew. His entire ground crew was not killed off by the Borg.Therefore First Contact represents a change in the timeline. The reason why the NX-01 was named Enterprise was because Deena told Cochrane the name of the ship through the telescope. It would be logical to assume that Cochrane wanted the first NX ship to be named Enterprise as he worked on the NX project. In the original timeline there was obviously no NX shipped named Enterprise. Anyhow what I am saying is that the original timeline was erased by the Borgs Time Incursion into the past and established a new timeline. Hence, ENT was born. I am looking at in-universe events not saying that Berman and Bragga intentionally wanted to create an alternate timeline like JJ Abrams. Although I believe I did read that Bragga stated something like that after the first season of ENT and only in the later seasons he went back on that and tried to link ENT with TOS more closely.
Even in this interpretation (which is definitely feasible) the only difference in the Augment story would be literally the name of the vessel doing the adventuring; thats it.....