Assuming Bush isn't forced to resign as well, I'd think that Dukakis would win, especially since Bush wouldn't have Reagan making his dramatic "Gipper" speech at the convention; remember, before the convention, Dukakis was up by 15 points in the polls.
On another note, Bush came in 3rd in IA, behind Dole & Pat Robertson, & for a time, it looked like Dole would beat him in NH too, so if he were potentially even less popular in this scenario, then Bush could very well lose the nomination to Dole, or at least get embroiled in a hard-fought, nasty primary that leaves him further weakened than he was in real life.
Yeah I have serious doubts Bush would be the Republican nominee in this scenario. Ford's hands were relatively clean of Watergate and he still barely won the nomination after Nixon resigned (although Watergate wasn't the entire reason for that). In this case, Bush would probably be implicated to about the same degree as Reagan. He would be highly tainted as a result so someone else, likely Dole, becomes the nominee. I'm not even sure Bush runs. And whoever the nominee is very likely loses, for a number of reasons.
First and foremost, after TWO Republican presidents (who both won re-election landslides) have had to resign due to scandal, hopefully the country would now find it very difficult to trust the GOP again. Second of all, probably Lee Atwater isn't slinging mud because the GOP's only hope now is to run a clean campaign and distance themselves from scandal and dirty tricks. If they throw stones at the Dems, they could very easily be thrown back at them and hit harder. For that reason most attacks they did try likely wouldn't stick and many would backfire. Maybe even Hart doesn't drop out because his "affair" either isn't exposed at all (some have claimed Atwater set him up) or isn't made into such a big issue. Regardless of who the Dem nominee is, he probably comfortably wins. As you said, at one point Dukakis was up big as it was. With a GOP tainted by scandal and neutered in its ability to attack effectively, they would have no chance.
What would be especially funny is if Carter won on a "Miss Me Yet?" kind of comeback campaign. Twice America screwed up with Republican crooks, twice they came crawling to Carter to set things right...
Really though, it was the right move for the GOP if not for the country to ride out Iran-Contra. I still can't believe the defense was “A few months ago I told the American people I did not trade arms for hostages. My heart and my best intentions tell me that's true, but the facts and evidence tell me it is not.” Let's be real, only Reagan could get away with that nonsense.
