Yes, Wolverine still ends up getting his adamantium claws.
They can be seen in one of the recent trailers for X-Men: Apocalypse, which takes place 10 years after the past-events of Days of Future Past.

Simon Kinberg explains to Cinemablend that life time finds a way, and Wolverine still somehow ends up working with Stryker and getting his claws, but that the hows-and-whys will be left unexplained.
One of the things we talk about in Days of Future Past - there’s a scene where Hank/Beast talks about the immutability of time. Basically, you can change the way the future goes, but it finds its way back. So while we don’t show how, it found its way back to Wolverine being part of Weapon X and under the thumb of Stryker, that’s the idea.