For those unfamilar with how Time Travel workd in the comic and how it's working in the movie check below...
In the Dystopian future the remaining Mutant rebels send Kitty pride back in time to warn the past X-Men about the dystopian future where mutantkind and mankind are under the control of the Sentinel robots.
She doesn't physically travel through time, Rachel Summers uses her powers to transfer Kitty prides future mind back into her past body.
Rachel Summers: "LET YOUR MIND BECOME ONE WITH ME. OUR MINDS ARE ONE WITH ALL"
Then the comic panel read 'Her Soul is flung across the abyss of Eternity'
In the present day (which in the movie will be 1973) kitty collapses and a panel reads...
'Suddenly reality twists inside-out, she comes face-to-face with herself, an older, sadder, wiser self -- And then her soul too, is flung over out over the abyss of Eternity'
When she wakes it's the future Kitty in her past body.
Kitty Pryde has not yet been born in the Movie-Verse in 1973 so they can't use her as the time travel subject.
They need someone who is physically in the past and available in the future portion of the movie.
There are only three candidates: Charles Xavier, Magneto & Wolverine.
Since Xavier and Magneto are a part of the unfolding events of 1973 which shape the future they can't be hijacked by their future selves as they are needed to change events.
This leaves Wolverine who not only is available in the time-period but also hasn't fully met Xavier or Magneto yet (the cameo in XFC was done so quick and fast it was fan service) and most importantly isn't an X-Man shaping the events, he is an outsider at that point, that frees him up to help change events.
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Right now it's muddled and unknown just how Wolverines conciousness will be sent back in time. Since Rachel Summers, a telepath, was able to do it in the comic its feasable that Charles Xavier, also a telepath can do the same in the movie.
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Wolverine in his yellow/blue?? where?
This pic