Up until the "modern era" of Trek, my understanding of canon is that the Eugenics War happened in the 1990s and was immediately followed by WW3. It seems like that has changed.
Picard season 2 took place in the 2020s, and there was no evidence of widespread devastation that would have taken place in a major war, although it could be argued that it wasn't shown to us on-screen. At the end of the series, we see a folder labeled "Project Kahn," hinting that either Kahn has yet to be born or that the season's "bad guy" is planning to continue Kahn's legacy.
In Strange New Worlds Season 2, Episode 3, the focus is on Kahn and how he affects the timeline. We see Kahn as a child, meaning the Eugenics War has yet to take place, even though the episode takes place in what is obviously 2020s or 2030s Toronto. Even more telling is that the Romulan spy says "This should have happened in 1993." Might have been 1992; still, early 1990s.
This leaves me with a few questions. If the Federation time-cops are so set on preserving the timeline, why did they allow the timeline to be altered to such a degree that the Eugenics War, a major event in Humanity's history, happened at least 30 years after it was supposed to. In addition, what event actually altered the timeline so that Kahn was born/created decades later than he should have been? As far as I remember, we haven't seen anything to show why this happened.
Since we now know from SNW that the Eugenics War happened some time after the 2020s, how does that fit into the timeline for Zefram Cochrane's first warp flight? Assuming the Eugenics War is still followed by WW3, that only leaves a max of 49 years between the start of 2 major conflicts and first contact with the Vulcans. In "First Contact," Cochrane and company assumed the Borg attack was from a hostile force on Earth. Perhaps WW3 was still in progress, and the events of 2069 were what ended the war?
On a side note, the destruction of a single ship spawned the Kelvin timeline. Since SNW shows us that events in Earth's history no longer match up with the timeline established in TOS, TNG, VOY, and DS9, does this mean that SNW (and possibly PIC) are also in a non-prime timeline?
For what it's worth, I wouldn't characterize them as "allowing" the timeline to be altered. They seem to be hard at work fighting to keep it as intact as possible.
As far as being a "separate timeline", it depends on where you draw the line. Being strict about it, one could argue that every single instance of time travel creates a separate timeline, so we witness dozens of assorted timelines throughout the shows, starting all the way back in the first season of TOS. Generally, however, we tend to refer to timelines as distinct only if there are substantial differences in the course of history, and "substantial" is subjective. In-universe, there definitely seems to be a certain degree of "good enough" when it comes to "restoring the timeline", even if minor changes persist. Until we see a substantial contradiction of events between SNW/PIC and TOS/TNG/etc, I'd say it's premature to declare them completely separate timelines.
(And, from a production perspective, it certainly seems like the intent is that SNW and PIC are "prime timeline". SNW just had a time travel crossover with LDS with no apparent references to crossing into an alternate timeline, and PIC S3 was filled with direct references to the events of TNG.)