In X-Men Origins: Wolverine, "Weapon X" is created and Wolverine escapes. In Apocalypse he is back in a case, and escapes again. How are these two Wolverine's connected?
Because of Logan's interference in X-Men: Days of Future Past, the original X-Men timeline has been completely changed. That means that pretty much everything that happens in X-Men Origins: Wolverine after Logan meets Stryker no longer happened the way it was shown in that movie.
In Days of Future Past, which takes places in the 1970's, Logan's is left unconscious when his future self's mind is retrieved. The last scene in that movie shows him being recovered by Mystique, posing as Stryker.
In Apocalypse, which takes place in the 1980's, we see that sometime in the decade between the movies, Logan has been captured by the real Stryker, and taken to the Weapon X program. This is how we find him, caged up and waiting to be released by the X-Tots.
We have no idea what happened to him in the interim. As described in this question he still, obviously, gets his adamantium, but the new movie timeline never shows that event happening.