It's canonically established that Star Trek: Enterprise is set within the Prime Universe rather than the Kelvin Timeline, but let me explain why the show technically belongs in both. The Kelvin Timeline debuted in the 2009 Star Trek reboot movie that was directed by J.J. Abrams and starred Chris Pine as James T. Kirk. Based on Star Trek: The Original Series, the movie replaced every member of the cast with new actors.
However, the movie's biggest twist revealed that it wasn't pulling off a traditional reboot. Instead, every other Star Trek TV show and movie that preceded it remained part of the main canon, only in another timeline. The return of Leonard Nimoy as Spock Prime alongside Zachary Quinto's new version of the legendary Vulcan confirmed this. The Kelvin Timeline has been officially revisited twice more on the big screen, but I'm convinced it actually showed up for the first time eight years before Abrams' movie.
How the Kelvin Timeline came into existence in the Star Trek franchise
As the name implies, the Kelvin Timeline branched from the Prime Universe when the USS Kelvin was destroyed at the beginning of 2009's Star Trek. Before that point, the two realities can be assumed to be identical. The Romulans who took out the Kelvin came from the Prime Universe's future, and the Starfleet ship's destruction resulted in things unfolding differently from how they originally happened. That said, the Prime Universe retained its integrity, and the Kelvin wasn't destroyed in the franchise's primary reality, allowing it to continue as we've always known it.
Key changes in the Kelvin Timeline involved how George Kirk (Chris Hemsworth) died, James T. Kirk (Chris Pine) taking command of the USS Enterprise under very different circumstances, and the interaction of Spock Prime with Quinto's version of the character. It is a very cerebral way of rebooting a franchise, but also very respectful to the space opera's storied history.
Enterprise is the only Star Trek show set before the Kelvin's destruction
There is some crossover between Star Trek: The Original Series and the first Kelvin Timeline movie, but not much. For example, Kirk and Spock are already born by the time of the USS Kelvin's destruction. Almost every other Star Trek movie and show is too far out of reach for them to be caught up in any way by the Kelvin Timeline. There is, however, Star Trek: Enterprise, which is set a century before TOS.
Any reference to TOS in Enterprise is always, at best, a token gesture. The two properties are set too far apart to properly interact, so it also makes it impossible for Enterprise to interact with the 2009 movie. As a result, the Kelvin Timeline and the Prime Universe are one and the same during Star Trek: Enterprise. The Kelvin hasn't even been built, let alone destroyed, so the alternate reality hasn't yet branched off to create the world in which the rebooted movies are set.
What this means for Enterprise's characters in the Kelvin Timeline
Although there are currently no solid plans to reboot other Star Trek eras like Abrams did for The Original Series, it's fun to think about that possibility. For example, a Next Generation-era Kelvin Timeline story would likely be very different from the original show due to the temporal changes, and heavy recasting that would be necessary for it to work.
For Star Trek: Enterprise, that wouldn't really make any sense. Because the show is set before the branch point of the USS Kelvin going up in flames, Enterprise's events are set in the Prime Universe when it was also the Kelvin Timeline, or to put it another way, before the Kelvin Timeline had been brought into existence. So, a throwback "reboot" movie of this era would just be every season of Star Trek: Enterprise, exactly how they already happened.
It should be noted that Admiral Archer, who was played during his captain era by Scott Bakula in Enterprise, does receive a mention in the 2009 movie. So, he was at least still alive in the time between the Kelvin's destruction and the time Simon Pegg's Scotty mentions the commanding officer of Star Trek: Enterprise's titular vessel. If Archer had been shown in the 2009 Star Trek movie, he would surely still have needed to be played by Scott Bakula.