History will not catch up to Star Trek because Star Trek is a work of fiction and some of it relies on violating basic laws of physics.

This is like someone in the 1930s asking what should happen when history catches up with the H.G. Wells-scripted film Things to Come.

The answer is: Nothing, and part of that reason is that we don't shoot people into space with a big cannon.

Roddenberry himself was adamant that Star Trek’s history had to remain a possible history for viewers. So, the dates can slip as long as the major events don’t.

That is why he put WW3 later than implied by TOS, delaying it to the mid 21st century in the TNG pilot ‘Encounter at Farpoint’ even though that led to a contingent of TOS fans insisting that it ‘had to be a separate universe from the one of the original series.’

While writers never explicitly resolved this onscreen during the Berman Era shows, preferring to weasel with offscreen head canon in interviews saying that perhaps the Eugenics Wars were covert and going on unknown in the 90s, the new shows have dealt with this problem head on by acknowledging that temporal incursions do affect the timing of major events without making it a separate timeline.

SNW and Prodigy have been able to make this clear onscreen in canon with the expert help of the franchise’s excellent physicist science advisor Dr. Erin Macdonald. (She did her PhD with the team in Scotland that got the Noble prize just a couple of years later. She’s truly on top of modern theoretical physics.)

Nothing - it's a TV show and we should consider it a huge What if? Not trying to match our history/timeline to the IPs history/timeline.

Besides our history is starting to look worse than the Bell Riots. Just ask California governor. He may not have created a Sanctuary in San Fransisco but he is criminalizing homelessness.

"But it's set 800 years in the past."

"Discovery isn't! although the less said about that one, the better."