The "Prime" timeline was the one where Ambassador Spock and Nero came from in the 2009 movie. But that is not necessarily the same timeline in which TOS took place.
After the Borg attack on Cochrane in "First Contact," and the Temporal Cold War throughout "Enterprise," with the Suliban attack on the Klingon and the Xindi attack on Earth, the Federation of the 2250s could look wildly different from TOS -- even BEFORE Nero showed up and destroyed the U.S.S. Kelvin. The "Prime" timeline is the result of 200 years of time travel shenanigans, with multiple aliens fiddling with Earth's history.
Also, when Admiral Janeway went back in time in "Endgame" and got the Voyager back to Earth decades earlier, she created a NEW timeline (which was the setting for "Star Trek: Nemesis," where Admiral Janeway was seen already back on Earth). Presumably, this alternate Janeway-verse is the so-called "Prime" timeline from which Spock-Prime originated.
Going further back, the original timeline shown in "Yesterday's Enterprise" showed different Bridge consoles and uniform designs aboard the Enterprise-D. Then Lt. Yar decided to go back in time and change her own past (exactly what Admiral Janeway and Nero did), creating the alternate TNG timeline, which had different ship designs and uniforms from the timeline Yar came from.
So there are multiple precedents for time travel into the past changing uniforms and ship designs, throughout multiple timelines in "Star Trek's" 50-year history. The "Prime" timeline (presumably the same one created by Admiral Janeway) and the "Kelvin" timeline (created by Nero and Spock-Prime) are just the last two of dozens of alternate timelines depicted over the past 700 episodes.
(The only actual canon depictions of the "Prime" timeline are in "Endgame" and "Star Trek: Nemesis," as well as Spock-Prime's flashback scenes in 2009's "Star Trek." Spock-Prime is not even the same Ambassador Spock last seen in TNG's "Unification" -- Spock-Prime is from the alternate timeline created by Admiral Janeway, while the original Spock may have died on Romulus years before the Voyager ever got back to Earth. And that Ambassador Spock is still not the "original" Spock from TOS, who may have died in the Klingon-Federation war of the original timeline from which Yar came in "Yesterday's Enterprise.")
Then there was the "Voyager" episode "Future's End," in which Henry Starling found future technology from a crashed timeship, altering much of the late 20th century's history. Maybe in this timeline, as a result of Starling's futuristic "inventions," the Eugenics Wars never happened, or happened differently from the TOS timeline; or maybe Khan became a blue-eyed, pale British guy instead of an Asian Sikh warrior with a Mexican accent. Who knows.
My point is, the so-called "Prime" timeline is about 20 alternate timelines removed from the "original" timeline depicted in TOS. As far as we know, everything shown in the "Star Trek: Discovery" trailer is exactly how the "Prime" timeline should have looked at that time period, since we have never seen the "Prime" timeline in the 23rd century depicted before.