This would depend on how realistic you'd want to be. In reality, by the late 24th century in the Abramsverse, things would have changed so fundamentally from the prime universe that the time period TNG would be in would bear little resemblance to how we know it.
But if what you're really asking here is, how would the prime universe TNG characters act like if they were in the Abramsverse but still somehow together on a nuEnterprise-D in the same setting as TNG, then I'd have to say that they'd act exactly the way they act now, only with Vulcan destroyed 100 years before, and a nuEnterprise-shaped model on the Enterprise History Wall instead of the original Constitution class. And probably no Dr. Selar.
But it wouldn't need to be set in the Abrams alternate
universe to be rebooted. The original timeline is already rebooted at the start of ST XI.
It might be interesting to see a series or film set in a rebooted prime universe around the time of the destruction of Romulus. Alternatively, it's easy to imagine a rebooted TNG crew establishing contact with the Abramsverse and striving to avert the destruction of Romulus... (Abramsverse would still exist seeing as we're now working on a multi-timeline model - I think).