We had 2 time lines in comic story, but Singer said we'll see few time lines, so at least 3.
IMO it will be as someone mentioned:
1. Future (2023) -few minutes at the begin of movie,
rest of movie: 2. Present (2013)/3. Past (1963)
and at the end of movie we will see 4. Alternate present, maybe AOA.
When Logan backs to present, it should be shown as in Butterfly Effect, what was changed after 1973 we should see in his mind, then he open eyes and sees new reality in 2013, Apocalypse rules
Maybe Bishop uses travel machine created by Forge to travel to present, or Blink sends Bishop to present using her portals. We don't have time machines in present time, so Bishop stucks here, and Xavier with Eric decide to travel Wolverine's mind to 1973.
If there isn't future time line in this movie, then Logan's grey hair maybe are affect of him seeing Xavier is still alive. All people are dying near him, but someone is back to life, and that's big shock for Logan and his hair become grey.