My gut plonks for between 2024 to 2025. This handwaves a few issues:

1. Why didn't the Travelers intervene? Talinn was dead by mid-2024, and in any case, it's possible that since they saw the DTI was already involved, they allowed the humans to work out the problem for themselves.

2. The production art in "In a Mirror Darkly" (which is shaky canon at best, I admit, given other dates in that same art which have since been retconned) puts 2026 as a start date for WW III, prompted by issues over genetic enhancement and involving Col Green. It's not a stretch to say that if the existence of the Augment children became public, it could trigger this.

3. Khan's age. Ricardo Montalban was about 46 when "Space Seed" was broadcast, and Khan did look like he was in his late 30s to early 40s then - which is plausible for a ruler of about 1/4 of the Earth for a period of 4-5 years. Our current date for the end of WW III is 2053, Riker saying that was 10 years before First Contact in 2063. So the year that Khan left Earth can be no later than 2054 or thereabouts. If we work backwards, then Khan would have been born around 2014. The young Khan we see in this episode looks to be about 10 years old, so our window becomes about 2024-2025, and I go for the later date because that would give Green a little bit of a runway before blowing things up.