Does Wolverine still have his adamantium claws after the change in events?

Yes, he still ends up with Adamantium claws (and a broken memory).

With the release of X-Men: Apocalypse we definitely know that Wolverine still ends up with Adamantium claws even in the changed timeline.

It is true that it was actually Mystique who picked him up from the river at the end of Days of Future Past and he thus had a much lower chance to end up with Stryker. However, during the events of Apocalypse, which are set 10 years later in 1983 in the already changed timeline, some of the (not yet) X-Men end up in a secret facility of Stryker's. There they eventually come across a cage that contains a so-called "Weapon X", which turns out to be noone else than Wolverine himself, full with metal claws and an apparently heavily damaged memory.

It is still unclear how he ended up with Stryker after he was recovered by Mystique, but there is no doubt that he did and that this facility (apparently in a snowy area) is probably the exact facility at Alkali Lake where Logan returns to in X2. And seeing his deranged memory and animal-like behaviour, his actual backstory as depicted in the first few X-Men films does't really seem to have changed much.

And while the filmmakers said that the events of X-Men: The Last Stand and X-Men Origins: Wolverine are largely eradicated from continuity, it is still left unclear how much of the first two X-Men films stays intact. In an interview with Collider directors/producers Bryan Singer and Simon Kinberg explain that while the specific events are largely open now, there is nothing that prevents them from panning out similar to how they did in the existing films either. And in this CinemaBlend article Simon Kinberg explicitly said that Wolverine's fate as "Weapon X" stays largely the same due to a certain immutability of the timeline with respect to specific key events (call it "destiny", if you want):

Borrowing a bit of scientific theory and dialogue from the last X-Men movie, Kinberg explained that the idea is that Wolverine was always going to end up in the Weapon X program despite all of the events that changed in his life due to time-travel antics. Said Kinberg,

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.

I wouldn't think so because;

1. the writers at Marvel (presumably) understand how time travel works. Meaning if Wolverine wasn't saved by Stryker in the movie and ends up not having Adamantium claws that would mean that all of the Wolverine movies as well as the first three X Men movies have an inconsistency.

2. He could meet Stryker again in the future/past(?) which would mean he would get the claws.

3. Hopefully it will be explained in the next movie (I haven't read the comics).

This probably wouldn't change the continuity of wolverine's past at all as I believe Mystique only uses the disguise to order some soldiers to fish his body out of the river.

Her reasoning for this could be that she realizes how important wolverine is in the possible future or far more likely she did it as a favor called in by xavier.