Logan in 1973 (X-men Days of Future Past) managed to set a new future for himself and for everyone else, which suggests that he may well have never had the chance to meet the real Stryker (again)...
There's also the possibility that Stryker didn't find his special "meteor fragment" in order to produce adamantium to inject in Logan's bones.
If so, Logan probably didn't have the chance to meet his girlfriend Kayla (X-men Origins: Wolverine) who told him about the sad story of Kuekuatsu (or Cuckoocachoo, as Logan promptly understood and identified later as the Wolverine)...
Therefore, how likely is Logan to become the Wolverine after 1973?
P.S.: Concerning X-men: Apocalypse
I haven't seen this movie yet, but I'd like to understand if Logan is the Wolverine in it. If he is, is the movie inserted in the older continuity?