That all depends on her powers actualy working as she claims, for which we have no actual evidence.
It's suspiciously convenient. She claims she reboots the universe (to the point where she was born) everytime she dies, yet if she would ever be asked for proof or physical evidence she would deny being able to provide any, because she claims any such evidence has been eradicated.
But everyone has to believe her because she has "memories" of such events, in a universe notorious for time travel, memory manipulation and super natural forces which mess with reality itself to suit their plans.
Just before the start of HoX/PoX, the X-men just had just been trapped in a pocket dimension where a crazed reality warper created an alternate history to suit his desire and even created fictional versions of people not stuck in said dimension, to manipulate the people in it. So why should we not at least suspect Moira X could be something similar?
Meanwhile if we look at the multiverse we see no other Moira MacTaggert who has even a hint of having such powers as mutant (i remember there were some where she was or became a mutant). But we have several who worked with Charles Xavier or otherwise became involved in events involving mutants.
Both which contradict Moira X's own recounting of her previous lifes in which it was established that only through becomming a mutant and living alternate lifes was she actualy motivated to get involved with mutants, Xavier and co.
So we have much more actual evidence speaking against Moira X's claim, than anything which proves them to be right.
Hence why i would say it's a bit futile to think about how her claimed powers would affect time travelers, especialy since the presence of time travelers can be another evidence speaking against Moira X's claims being true.