The "new" continuity that exists after Days of Future Past isn't actually clear. We know that the movies X-Men Origins: Wolverine and X-Men 3: The Final Stand are no longer canon, but that doesn't necessarily mean that those events didn't happen in some form or another.

In particular, note that all of the remaining X-Men films are still canon, so all of those things supposedly still happened. That includes, for example, X-Men where Wolverine's adamantium bones are a significant plot element. So obviously, he must have gotten them somewhere. Perhaps we'll learn where in the next movie, or perhaps in the upcoming solo movie.

At the end of Days of Future Past in the past timeline, when Logan's future mind leaves his past body, we see him being retrieved from the river by Mystique, disguised as William Stryker. According to the director, this is supposed to represent Mystique saving mutants from their original fates, but in truth we have no idea what happened to him after that point.

Also, note that the things that happened in X-Men Origins: Wolverine prior to 1973 should also "still have happened", just perhaps not as they were shown in Origins. But Wolverine first met Stryker and joined Weapon X either during or shortly after the Vietnam War, which effectively ended in 1973 as well. It's never explained why Logan isn't in Vietnam during Days of Future Past, unfortunately, but it's not unreasonable to assume that Stryker still has his sights set on Wolverine -- which would explain why Mystique was impersonating Stryker in the end scene -- and he will still end up giving him the adamantium sometime between the 1970s and the 1980s.

Because Stryker just implanted the adamantium in him.

In X2 and X-Men Origins we learn all about how Stryker implanted the adamantium in Wolverine at Alkali Lake some time in Wolverine's past. This scene in X-Men: Apocalypse, where Stryker attempts to imprison the other mutants at Alkali Lake represents that event in the altered timeline.