Regarding the Abrams movies, this is an ongoing conversation, because the only evidence that they are NOT part of the prime universe are off camera interviews, which I will not consider canon. I call this a rebuttable presumption, because if it is truly the intent to establish the prime and Kelvin timelines as different universes, they can always demonstrate that on screen in a future movie.
The Abrams time travel attitude, while valid for him, does not reconcile with how time travel works in Star Trek, so given that Abrams didn't create Star Trek, the Roddenberry method prevails.
For me, that means that the Kelvin timeline overwrote the prime timeline, with Spock Prime, who just happened to be one of the time travelers, being the sole survivor after ST09, and now, no longer.
There is no dialogue to disprove that. The closest is Uhura talking about an alternate reality, but even if the timeline is overwritten, that's an alternate reality. Abrams chose to be ambiguous, so I consider the decades of Star Trek before him to override it, and nothing in canon disproves it.
As for Discovery, THAT could be an alternate universe. There is no connection between Discovery and the Kelvin Timeline.
If you buy into the idea that Kelvin is separate, I still don't consider Discovery part of the prime timeline, since very little about it seems to work with it. Even the Enterprise doesn't look the same.