First off, shouldn't this be in the After 1900 forum rather than Future History?
Second, ooh, a horror movie WI. Me likey.
Reading the Wiki page for Terminator 2, the idea of making a sequel emerged the moment the first film became a hit, but disputes between Hemdale Film and Carolco Pictures over who owned the intellectual property rights held up the sequel for several years. It only ended after Carolco head Mario Kassar, at the urging of Arnold Schwarzenegger, paid Hemdale $5 million for the rights in 1990. Perhaps the dispute is resolved much earlier, before the development of the CGI technology needed to bring the T-1000 to life. As a result, either Carolco or Hemdale (whichever studio owns the rights) puts a sequel into production sometime in the mid-late '80s with a heavily scaled-back script that plays out more like the first film.
(Edit: and possibly more like a slasher version of OTL's Terminator 3, with a Terminator sent back in time to kill John Connor's generals and lieutenants. Thanks for the suggestion, ColeMercury and Tyr!)
Another way to do it would be to have James Cameron's film The Abyss fall apart. IOTL, that was the film that helped resolve the technical concerns that many had with the CGI technology needed for Terminator 2. It was also a spectacularly troubled shoot, with Cameron -- no stranger to troubled productions -- calling it the worst film he's ever worked on. 70-hour work weeks for six months, one of the giant water tanks (built inside an unused nuclear power plant cooling tower) springing a leak and needing dam-repair experts to fix, and most notoriously, both star Ed Harris and Cameron himself nearly drowning at separate points in the production. Have a few more things go wrong -- maybe Harris or another cast member actually drowns on set, maybe the water tank springs another huge leak or two -- and you could turn The Abyss into the Heaven's Gate of the summer blockbuster era. Studios dramatically rein in their budgets after this disastrous shoot, and Cameron is in career purgatory. Carolco still gets the Terminator rights as in OTL, but with the failure of The Abyss and Cameron's disgrace still fresh in everyone's mind, they go the low-budget route, scaling back the script and hiring a journeyman director to make the film.
Now that you've turned Terminator 2 into a sci-fi/slasher horror film much like the first one rather than the SFX-driven action blockbuster it was IOTL, you have to make it successful enough to spawn a franchise like the OP is asking for. Now this is the tricky part. Under the first scenario, you'd see the alt-T2 released sometime in the second half of the '80s depending on when the rights got sorted out, by which point the slasher genre was at the peak of its popularity. Any future sequels, even if they'd been rushed out the door, would've been made in the twilight years of the '80s slasher boom. So unless you get the third film out the door by 1988 and get yourself a neat trilogy, any future films would find themselves in the same place as such "classics" as Freddy's Revenge: The Final Nightmare and Friday the 13th Part VIII: Jason Takes Manhattan -- even if they're good, nobody will be going to see them.
The second scenario -- the Abyss scenario outlined above -- is even more difficult, since in that case the film would be released in the early '90s, by which point the horror genre was in a doldrum that it wouldn't get out of until Scream. You could still make a good, low-budget T2 at this point, but it would look a lot more like Die Hard than Friday the 13th. Not that that's a bad thing, mind you, but it's not what the OP is requesting.
