With the ending of X-Men: Days of Future Past, Logan was given a glimpse of a time altered present day in which Jean Grey (Phoenix) was still alive, as was Scott Summers (Cyclops).
With this change in the continuum of the X-Men timeline, Logan (Wolverine) had stayed with the X-Men since 1973 and reopened, with Professor Charles Xavier and Hank McCoy, The School for Gifted Youngsters. As a member of the faculty, Logan was The Historical Adviser to the students, having firsthand knowledge of the past 160 years of history.
The fact that Logan is part of the team from 1973 onward changes a major dynamic in his relationships to the other team members and students, especially the younger versions of Storm, Jean Grey and Cyclops.
1. Jean Grey
When Logan first meets Jean she is probably 13 years old (circa 1980ish?) and there is no way he will still have romantic feelings for this young girl - the future woman he loved (and killed) in the old timeline. It would just be too creepy in that sort of Roman Polanski type of way. Even in 1980, Logan would still look like he does now: A man in his mid-forties. His relationship with her will be more that of a father figure. By the time we see Jean (in 1985ish?) she would be just around 19-years-old.
2. Scott Summers
Much like with Jean, Scott will be a teenager when he first meets Wolverine. The whole rivalry and love triangle friction between Scott, Jean and Logan will be totally gone. Logan will be seen also as a mentor/father figure to Scott, too. It will be interesting to see if Scott's brother (Havoc) will have a part in this movie which will be set in the '80s.
3. Ororo Monroe
Wolverine's relationship with Storm was rocky at best. She was often scolding him for not taking sides in the X-Men's fight with Magneto. She was also disapproving of Logan's feelings for Jean, But since most of that won't really happen now, they may have a better relationship. Storm eventually leads the X-Men (in X-Men: The Last Stand) but that is all retconned now.
4. Old Friends, Enemies... Frenemies!
Much of the adversarial nature to the relationship between Eric and Charles should be gone. Magneto's relationship with the other X-Men (except for Wolverine and Beast) will be changed for the better as well. The events of X1 & X2 - or maybe even X3 for that matter - will most likely not happen.
Mystique may be the new adversary of both Charles and Eric, after her foiled attempt to kill Bolivar Trask, she has made herself the enemy of both of them.
5. To Weapon X or not to Weapon X? That is the Question.
The Wolverine of 1973 still had his bone claws. It is unclear if after Days of Future Past whether or not he will still undergo the Weapon X procedure that made him indestructible or whether he will avoid William Striker altogether. Apocalypse may be the one to put Adamantium into Wolverine...if he does indeed turn Logan into the Horseman of Death!