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?
I feel like, to answer why the federation time-cops are letting this slide is because fundamentally, it doesn't change much, the events still happen and affect the current galactic-chrono-political thing they have going on, and if they do interfere they could be facing an all-out time-war/"Temporal Hot War" with factions like the Sphere-Builders, Romulans, Na'Kuhl, The Mysterious Benefactor of the Suliban, and other factions that have not been revealed. So I'm willing to think that they are letting that slide so that the timeline isn't damaged irreparably. And they do state that the Temporal Cold War and its factions are trying to eliminate Khan since he's the focal point of the Federation's creation (and kind of what separates our timeline from Star Trek's), they've pushed back Khan 30 years. She does say that she arrived in 1992 and was waiting 30 years to kill him.
To answer your second question and side note, those events will still fit. Just because Khan was moved up doesn't change the whole timeline. To be fair, in "Past Tense" and "Future's End", those cities don't exactly look like they've been ravaged by war, so it's more likely TNG, VOY, DS9, and SNW are their own timeline rather than TOS, TNG, VOY, and DS9. Plus, 49 years is more than enough time to have 2 wars in, especially if they happened back to back.
And, to be fair, Star Trek isn't exactly a strict timeline. We see some differing events all over, probably because Star Trek likes to give things dates. This doesn't necessarily mean it's a new timeline, just that the events happened other than what is said. Temporal Events in Star Trek are supposed to be taken with a grain of salt.