It can still kind of work. Montalban was about 45 when he was Khan, so let’s say Khan was around that age when he was exiled. The young Khan we see seems to be about 10 years old, maybe a bit younger.
So say baby Khan was born in 2012 if we want to take Sera’s 30 years literally rather than as an approximation. World War III (according to ENT: “In a Mirror Darkly” but the years may have slipped) starts in 2026 and lasts until 2053 (ST: FC, SNW: “Strange New Worlds”). Khan could easily have fought in the war and took power in the end days of the war - he’d only be 41 in 2053.
Even in the old timeline Khan only ruled one quarter of Earth for about 4-5 years between 1992 and 1996. So it’s not implausible that the Eugenics Wars happen around 2048-2053 (Khan would be in his mid-thirties, and augmented) and Khan escaped after his reign was toppled during the Last Day in 2053 on a non-warp powered sleeper ship, because Cochrane only managed warp 10 years later.
In fact, having the Eugenics Wars take place around 2050 works better because Archer said his great-grandfather fought in them (in North Africa). Since ENT takes place in the 2150s, that only makes about a century between their births, which is certainly reasonable, whereas if Archer-great-grand-pére fought in the 1990s then it’d be stretching his longevity just a tad.