This is in my to do list for a potential TL. I keep wondering if the TL is too soon.
Bush dies, and his legacy is that of a Republican version of JFK - a man who, faults aside, did what he felt was best for the nation, and elad us through some dark times. The Bush family becomes the Republican equivilent of the Kennedies - in my potential TL, Jeb Bush wins a landslide election in 2012.
Cheyney gets LBJ syndrome and gets it hard - maybe a few more of the things Bush pushed for get passed intially like Entitlement reform and a few other things, but the honeymoon period won't even last a year. The Republican Congressional scandals, Katrina, Abramov, the worst of the Iraqi insurgency will hit soon and hit hard, and unlike Dubya, he won't have Bush's 5 good years. He and the GOP will get raked over the coals and the country will suffer for it.
The Cheyney Presidency will look a lot like Bush's last three years, but in two major areas - gay rights and whomever he picks as VP. I had him pick Colin Powell personally, but whomever he picks will have an enormous impact on the GOP and the nation.
The Dems will clean up in 06 and 08 far more than OTL, but I don't see Obama getting the nomination. In 08, expect President Clinton - or if you want to go grimdark, President Edwards.
Pop Culture will also be effected, mostly for the better. There will be no glut of anti-Bush media (propaganda by any other name), and a lot of loud anti-Bush critics will get silenced or get smashed - think Dixie Chicks on steroids - think Green Day getting blacklisted, or for the V for Vendetta movie flopping. It will make pop culture a lot less political, which will be an inprovement in the long run I would think.
Overseas, we may see a bigger puch to get Georgia and Ukraine into NATO.
How does that sound?