D Wire Newman appeared to have been President from 1986 to 1990. Lassiter followed him & was a two term President & Bartlett's predecessor. Leo was apparently his Secretary of Labor despite being a Democrat.
I could easily see a Hoynes/McGarry presidency that was more on the moderate/conservative side. If Hoynes had to resign because of his affairs then Leo would have been the presumptive Democratic nominee in 2006 as his successor. If Leo still dies or doesn't run because of his health, or if Hoynes is impeached and doesn't resign a la Clinton, then Arnold Vinick most likely becomes the next President.
Actually in the show it is never stated WHEN Lassiter or Newman were in office. In the episode it is clear Lassiter is meant to be a Reagan stand-in, and Newman a Carter stand-in. Also the predecessor to Bartlet was a Republican. The show did reference that the Presidents were the same until at least Nixon but no more, Ford, Carter, and others were never mentioned in dialogue. There WAS a Ronald Reagan Hospital shown on the show but that's because they shot on location next to UCLA Ronald Reagan Medical Center.
A popular fan interpretation, and one followed by WW2010 and WW2018 threads here is that OTL presidents persist until 1986, then Bush (sorta), Newman (1 term), Lassiter (2 terms), Bartlet
The other leading interpretation is that West Wing follows a timeline where Nixon was impeached and new presidential election was called in 1974, with a Carl Albert being a caretaker President for a few months. Then Newman (1 term), Lassiter (2 terms, 1978-1986), Someone (one term 86-90), Bartlet's Predecessor (two term Republican, 1990-1998), then Bartlet
However going to the original premise, he'd likely be more of centrist, but it's only a matter of time before a scandal rocked him. Whether or not he held on to be re-elected is debatable. Josh might have found his way back into Hoynes' good graces, but I don't think any other regulars besides McGarry (who was a cabinet secretary under a Republican president) had a chance. Sam wouldn't have left that law firm, and CJ might wind up as Press Secretary still if McGarry is able to bring her onboard, but talk about awkward. Hoynes did have Senate experience but he would face a hostile congress.
The question is who Hoynes would choose as VP... could be "Wiley" who was big in the primaries alongside Bartlet, but I doubt Hoynes would accept Bartlet with MS as VP. So whoever Wiley is probably VP, doubt it would be McGarry, Hoynes needs more left muscle to balance the ticket