A Bush presidency post Nixon is certainly an interesting idea, I've been thinking about doing a TL where Bush is chosen in 1973 instead of Ford in fact.
I think Nixon still wins in 1968, though he may get slightly more muted conservative support due to Agnew's absence from the scene (from what I know of Agnew, he was a lot more popular than Nixon with the growing Conservative movement within the Republican party).
1972 is also simelar-Nixon still gets his lopsided victory and Watergate unravels soon afterwards.
I'm not sure if Agnew not being around speeds up the Watergate investigation-I get the impression that noone of either side of the political isle wanted an Agnew presidency, so with someone less corrupt waiting in the wings, things might move quicker.
Let's assume Nixon still resigns in the way he did, though. Does Bush pardon him? I strongly suspect he does, though he might not do the blanket pardon Ford did, which could help.
I think Bush is still primaried by Reagan, but Bush will win the primaries in the end.
Whether Bush loses depends on the campaign-with butterflies, Carter might not be the democratic nominee, though lets say he is. I think Bush is less gaf-prone than Ford, which could help him.
If he's been VP between 1969 and 1974 though, he might have a tougher time than Ford at disasociating himself with Nixon. Ford never actually faught an election with Nixon and only became Veep a year before Nixon resigned, which was after the Watergate activities took place.
I'd say on balance, Bush's chances of pulling off a win are about as good as Ford's were, so it could well happen. If it does, he may be able to handle the Foreign Polecy challenges Carter faced a lot better. Economic issues still harm the Republicans in 1980 though and Reagan (or whoever the GOP nominee is) loses to the Dems. I'm not sure who the Dem nominee will be if that happens, Mondale and Kennedy are the frontrunners, though that doesn't mean either of them end up being nominated (I suspect they wouldn't, actually).
If Carter wins, it'll be close like OTL and I imagine his time in office goes about the same. In this scenario, could Bush mount a significant challenge to Reagan in the primaries in 80? I'd imagine Reagan still ends up the nominee, but with Foreign polecy concerns at the forfront of many people's minds, it's not beyond the relms of possibility for Bush to upset the Reagan applecart so to speak.
If Reagan is still the nominee in 1980, who does he pick as his running mate? Might Former president Bush be more keen on the "dream tickit" idea than Former President Ford was in OTL? You might think that things from there go exactly the same if that happens, but not necesarily.
Until 1988 things will undoubtedly be very simelar, when presumerably Bush goes for the nomination again. With a previous presidencial record behind him, does he clinch it? I'd imagine he does-and he could well win like OTL.
I'd imagine Bush's term goes about the same if he still wins, but one difference will be the fact that Bush won't be able to run in 1992, so many democrats who didn't want to run against Bush at the hight of his popularity in OTL might decide to go for it.
Could we see Dole VS Cuomo in 1992?
If Bush doesn't end up as Reagan's running mate he may well still be in the cabenet, as Sec State or something. I wonder how that'd affect the Reagan administration.