So, leaving aside the side issue of the MU, my hunch on how to best carry forward the idea of an alternate timeline in a TV series is to continue the idea that the characters are preteraturally aware that their timeline has diverged from what it should be and are working to somehow set it right.
Which isn't my first choice or my hundreth choice for a premise for a new TV series. It's best to just leave it to Abrams to re-merge the timelines and if he fails to do so, the TV series shold ignore the issue entirely.
Otherwise it's just a big boring reset button. TOS was recast in the movies for its PR value but none of the other series have that awareness and the characters are not as iconic, so let's just have a new cast of characters.