Here's how it might have gone:
1979 - Star Trek The Motion Picture (no changes from OTL)
1982 - Star Trek The Wrath of Khan - Vice Admiral Kirk is Starfleet Academy Superintendent, Captain Spock is CO USS Enterprise, Commander Sulu is First Officer, takes command after the "death" of Captain Spock
1984 - Star Trek The Search for Spock. Sulu is promoted to Captain, given command of USS Republic, upgraded Constitution class. The rest of the original crew head back to Genesis.
1985 - Star Trek The Next Generation follows the exploits of Captain Sulu. Marc Alimino is Commander Tucker First Officer. Mark Hamill guest stars as Rear Admiral Stevens.
1988 - Star Trek The Voyage Home. Kirk is demoted to Captain and given new Enterprise. Commander Tucker dies at the end of 3rd season of Star Trek The Next Generation, Chekov becomes Republic's First Officer.
1991 - Star Trek V - The Undiscovered Country. Star Trek The Next Generation ends
1992 - Star Trek The New Voyages begins 60 years after the end of The Next Generation. Patrick Stewart as Captain Picard, Sheryl Lee Ralph as Commander Wickers, First Officer, Jonathan Frakes as Lt. Commander Riker, Chief Engineer, rest of the crew as OTL on the USS Enterprise.
1994 - Star Trek VI The Tomed Incident, includes original and TNG crews. Death of Captain Kirk, Sulu promoted to Admiral, Chekov and Uhura get own commands, Spock retires.
1997 - Star Trek Deep Space Nine begins
1999 - Star Trek New Voyages ends
2000 - Star Trek VII First Contact - Spock, Sulu, Chekov and Uhura reunite as the Federation is attacked by unknown race from outside the galaxy
2001 Star Trek Voyager begins
2003 - DS9 ends
2005 - Star Trek VIII - Devastation: The Battle for Bajor (return of New Voyages and DS9 casts)
2009 - Star Trek Enterprise set 50 years after DS9 ends. Stars Scott Bakula