We won't know the answer to this, unless/until we see it happen (again) in a future movie.

As you noted, Days of Future Past changed the timeline; that means everything we saw happening in the first three X-Men movies, plus X-Men Origins: Wolverine, is no longer necessarily true.

In the original timeline, as far as we know, Logan did not know anyone from Xavier's school until he was rescued by Cyclops and Storm. Though there's a strong implication that Xavier knows Logan, none of the other X-Men show any recognition. However, in the new timeline, we already know for sure that's not true -- he met several of the future X-Men, plus Mystique and Magneto, in the 1970's during Days of Future Past.

So, we can be pretty confident that Logan's introduction to the X-Men can't happen the same as it did in X-Men -- at least some of those people in the mansion will know him. It's also likely that he will, at least, remember Jean Gray.

In order to know for sure, we'll have to see it happen again on-screen. We know that Hugh Jackman has agreed to do at least one more X-Men movie, but the rumor is that it's going to be based on Old Man Logan, a story set well into Logan's future, so unless there are flashbacks that's unlikely to help. We also don't know what time frame the next X-Men sequel will be set in, to know if Logan will show up in the X Mansion during that film. Currently the plan seems to be at least 3 more movies (another trilogy), which would likely cover the right time frame, and it would be very surprising if Wolverine wasn't in them, but it's all speculation at this point.

But as we can see in the climax of Days of Future Past, Striker takes him, and we can see that he was kept captive in Apocalypse and young Jean Grey and Scott helps him, and he unknowingly helps them. So, they must recognize him when he showed up at the school, right?

DOFP does not show Stryker taking Logan in the Climax. In the Epilogue, we see Mystique rescue Wolverine from the Potomac River, having taken Stryker's form. This is 1973.

Apocalypse shows us that in 1983, a decade later, Wolverine has somehow ended up in Stryker's hands again.

In DOFP's "good ending", we see Logan waking up in 2035, stating he has no memory past drowning. He only remembers the old timeline, not the new timeline. Because of Kitty Pryde's power in DOFP, his mind has snapped back.

That means everything from 1973 to 2035 is a blank to new timeline 2035 Wolverine.

And because of the new timeline, where Jean is born at least 10 years earlier somehow, it's completely different than the old timeline.

X1 is 2000, while X3 is 2005, and the flashback shows 20 years earlier, Xavier and Magneto meet a young Jean for the first time. That puts it 1985. Yep.

So there is no way to know how they meet in the new, good timeline, at some point later than 1983 shown in Apocalypse.