From the perspective of the time traveler, he doesn't know he is in another universe. I live in 616. I go back in time and alter the history that I am aware of from 616. The act of my going back in time actually puts me in universe 617 but I have no way of knowing that as I time travel. So when I kill my grandfather I don't disappear because my grandfather in 616 still exists and I actually killed poor gramps in 617.
So the past and future Wolverines goes to and returns too were not the past and futures from his original timeline. And you could easily solve the issue of Wolverine already seeing the end of his new timeline at the end of DOFP with Apocalypse. In the many worlds interpretation, alternative universes are created all the time based on decisions we make irrespective of time travel. So if I am deciding whether to go to the store today or not, under the many worlds interpretation, that potential choice means that an alternate universe is created so that there is one universe where I decided to go to the store and another universe where I decided to stay home.
So there is more than one universe where Wolverine went back in time because subsequent to him doing so, other decisions made by other people within that universe would have caused other alternate universes (all with a past where Wolverine went back in time) even without time travel. So the end of DOFP is simply one of those universes while it's like DOFP would be set in another one of these universes where presumably Apocalypse woke up.
This would become confusing for an audience and kill the suspense as if we are just switching between different versions of a character that exists in different alternate universes, the audience can't really connect with any one character because we have such an individual concept of identity. So the easy solution from a story standpoint is to simply ignore the fact you are following the MWI as most audiences aren't going to care.