In one of those "what would you change about TNG" threads I've actually speculated about this scenario, setting TNG about 10 or 20 years after "The Undiscovered Country".
My idea was that one of the things the show would deal with was with the fallout resulting from the Praxis Crisis, the Klingon Empire would have broken up into several independent polities, with many worlds directly joining the Federation, most of the rest being associated with it, some of them getting subjugated by the Romulans and only a few Klingon worlds on the far edges of the old empire remaining independent, and often antagonistic to the Federation (that way the show can have both friendly and antagonistic Klingons) While all Klingons would have the updated makeup, there would be diversity in culture among them, with some resembling more the TOS/Movie Klingons in cultures, and others resembling the "spake Vikings" from TOS.
I'd keep the Enterprise being a Galaxy class and the general aesthetic of both the TNG ships and the uniforms, the changes from the Movie era aesthetic would be explained as a reaction to the Federation losing their biggest rival (the Klingons) and the Alpha Quadrant being perceived as more peaceful as a result. It wouldn't go as far as Early TNG Riker declaring that "Combat is only a minor part of being a Starfleet officer", but it would be more science/exploration focused.
Worf, in this continuity would still exist and be a member of the crew, but instead of a Klingon raised by humans he'd be the son of Klingon immigrants into the Federation, who came to Federation Space with his parents as a kid.
Troi would be changed to a telepathic human like Miranda Jones and, in fact, instead of Lwaxana we'd have Miranda Jones appear as Troi's mentor on occasion. She'd also be the science officer of the Enterprise instead of a "counsellor"
Like suggested above I'd also have more TOS aliens appear, alongside new ones. For example, I'd have the Ferengi and the Orions be in an alliance causing trouble for the Federation (that way the Ferengi can be the behind-the-scenes guys and the Orions can be their more active, more intimidating 'muscle', or alluring diplomats)
The rest of the characters would largely stay the same or have changes that aren't directly tied to the shift in time period.For instance I actually like the idea of casting Avery Brooks as Riker and changing the character accordingly, and also casting somebody else as Tasha/Marsha/Other Name and making her a much more prominent character. Data would definitely be compared to the various androids encountered in TOS and I'd possibly stick with the idea that he was built by an unknown civilization.
There could be an option for various characters, both major and one-off, from TOS to appear in cameos and minor roles. I'd might even go for a bit of small universe syndrome and have Picard state that he knew Kirk and worked with him to some extend when they were younger.
The whole "no interpersonal conflict or dark sides to the characters" stuff from the early TNG seasons would, of course also be nixxed, but that, again is not related to the shift in time period.
