This is a common trope in the speculative fiction genre. Time travel that alters the past allows the person or persons at the center of a shifting time stream to remember that the past has been altered.
Since Logan was the time traveler and the initiator of the events, only he will remember the previous timeline. For everyone else, they never happened. Is this a different universe? Did the other universe cease to exist? Did Logan's mind move to a different continuity? These questions are not clearly able to be answered without more information.
What we do know is:
In the case of Wolverine in Days of Future Past, he remembers what happens in the past that he went to but would have no recollection of what has transpired in his new timeline.
In his head, he would hold the memories of his old timeline and all the events that lead up to him traveling in time, his adventures in the past and his return to the present/future and his meeting with the Professor.
With Xavier's powers, he would be able to learn of the previous alternate past and update Logan with the new, altered history since Logan changed it.