Picard accepts an admiral position and he and Data go off on their own adventures to, Idk, further android rights or something. Before Picard leaves, he and Beverly finally become a couple and marry between seasons 7 and 8. Dr. Beverly Picard remains aboard the Enterprise and has a long distance, mostly off-screen marriage with Jean-Luc.
Promote Riker to captain, but have there be competition. Bring on Shelby as a contender to captain the ship, and the two of them butt heads a lot. Ultimately, of course, River becomes captain. I'd imagine he offers Shelby to stay as first officer but I would think she would decline, because she wants to be in charge *somewhere*. With her gone, I'd personally like to see Worf get promoted to first officer.
So...we need a new security chief and a new operations officer. OP says we're assuming DS9/VOY don't happen. I'm not that creative and it'd be meaningless to try and create new characters out of my own head for this thread. So I'm going to cherry pick some legacy characters to fill the gaps.
TNG Season 8 cast:
Captain William T. Riker
Lt. Commander Worf (first officer)
Dr. Beverly Picard
Lt. Commander Geordi LaForge (second officer)
Counselor Deanna Troi
Lt. Commander Tuvok (taking over security)
Lt. Jadzia Dax (science officer)
Lt. Sam Lavelle (operations)
Ensign Sean Hawk (comm officer)
Seven of Nine
Without Data, TV execs are gonna want something to bring in the ratings. How about a bigger focus on Worf and Klingons? Relations deteriorate and essentially we get the early Dominion v. Federation type hostilities but with the Klingons instead. The DS9 storyline with Kurn plays out on TNG instead. Jeremy Aster moves back to the Enterprise as Worf's adopted son and brother to Alexander. Jeremy because he was taken into the House of Mogh and never heard from again and I thought that was weird.
Anyway, the Romulans sit back to watch what happens after the dust settles. Maybe we get something akin to In the Pale Moonlight with Riker trying to bring the Romulans into the war. He negotiates with Sela, who fills the role of Vreenak...though she maybe or maybe not dies. Not sure who would fill the role of Garak in this scenario. No one on TNG was anywhere near as shady and complex who could make it work convincingly.
Hawk has a Rejoined type episode to feature the first male-male kiss in Star Trek. I guess the obvious choice from the list above would be with Lavelle but I don't remember if Lower Decks depicted his sexuality much. Doesn't matter really even if they did.
We get a lot of Trill stuff with Dax...also exploring her host's past with Klingons in light of the war and on a TNG setting.
Worf and Troi are still together but break-up mid season. Paving the way for Worf/Dax and Riker/Troi.
More Borg stuff, and the Enterprise crew encounters a drone (Seven) and liberates it from the collective. Jeremy becomes enamored with Seven the way Harry Kim was early on with Seven (though Jeremy is much more tolerable as a character than Kim ever was).
Probably all dumb ideas.