pragmatic conservative
Actually Cheney would probably be more openly pragmatic than GWB, although just as conservative.
To be honest, at first, like a lot of the other responses, I thought GWB was dancing to Cheney's tune. I have come to the conclusion that Bush is basically running the place, and doing a pretty wretched job, but this is (of course) a personal opinion.
What would happen? Two things, if GWB is assassinated as part of 9/11, you get an even greater national outcry, and a great deal of sympathy for the adminsitration.
The US would gear up and hunt down Osama, and, as in OTL, probably not find him.
It is an open question as to whether we (the US) would have attacked Iraq. I do not believe so, in that I believe there is a bit of 'I'll finish what my Dad started' in the whole thing.