As usual, the show does not clearly explain their time travel theory (likely on purpose, since any explanation is guaranteed to have holes in it). But in this case I think there's a simple answer:
That "other Barry", in the new timeline, still traveled back in time to get Thawne's help getting faster, he just "came back" a different Barry.
We have already seen cases where Barry is able to remember alternate timelines when no one else can; this is at least the fourth time he's traveled into the past (though he didn't always change things every time). We should probably just write off any such inconsistancies as "the Speed Force did it", since that excuse tends to work well in the comics.
As for everyone else, note that Cisco and Caitlyn were obviously waiting for Barry to come back forward in time, which means "their" Barry must have done the same thing "our" Barry did. They were also prepared for the time wraiths when he showed up. It's possible, for example, that "their" Barry decided to time travel because he knew he was supposed to to close the loop, or something similar.