The fact that the whole time travel is based on some arbitrary magic mutant power allows them to solve this potentially pradoxical issue in a rather convenient way, which is that Logan's changes to the timeline don't take effect on the future until he wakes up from Kitty's grasp.
This is explained by Kitty before she sends him back:
Basically, your body will go to sleep while your mind travels back in time.
As long as you're back there, past and present will continue to coexist. But once you wake up whatever you've done will take hold and become history. And for the rest of us, it will be the only history that we know. It will be like the last 50 years never happened. And this world, and this war, the only person who will remember it is you.
This is also later referenced by Xavier when Bobby wants to wake up Logan because Kitty is injured:
Bobby: She's lost too much blood. She can't keep going like this.
Erik: We have no choice, Bobby.
Bobby: We can wake him up.
Xavier: No! They need more time. I had a glimpse into the past. If we wake Logan now we may set our fate on an even darker course. They need more time.
This even suggests, that they don't have another chance to change things after waking up Logan. So their current timeline will actually vanish in favour of the new timeline once he wakes up and won't just go on in parallel to it.
So until Logan wakes up, none of the stuff he does in the past has any influence on the future. But at the very point where he ends his time travel it basically introduces an explicit synchronization point and the new timeline he set in motion will take effect, with the original timeline vanishing.