I think at some point in the first act, the leaders of both teams, Sam and Yelena/Bucky\* will recognize that because the world is quite possibly coming to an end, and they have bigger problems than the name dispute, their teams merge into the Avengers, co-led by Sam and Yelena/Bucky\*.


\*It's not confirmed who the leader of the New Avengers is, Jake Schreier said that Bucky was the 'de-facto leader' in press for the movie, but the Wiki says that Yelena is the leader, so I just have it as 'Yelena/Bucky'
How I think Doomsday will fix the Avengers/New Avengers problem