I very much doubt Gore would stay on as VP. He and Hillary never got along to begin with they always competed with each other for influence in the decision making. It reached critical mass in 2000 when Hillary decided to go for the Senate which infuriated Gore. In this world I don't think she would so there wouldn't be that between them but it wouldn't nice between them. Gore would probably want to go back to Tennessee and either get elected to office again or wait until 2004.
I think Bill would win the nomination no problem he'd have some challenges though; Bill Bradley at least. Maybe Al Sharpton perhaps but Bill wins easily besides in the Northeast. For VP he'd go 3 different ways: 1. Get someone with Washington experience (ie Gephart,) 2. Go for someone very progressive (ie Dean,) or 3. Stay in the south and pick Edwards. I'll go with Gephart for this officially.
Bush would pick Cheney more than likely which helps the base. Race for the summer would be a lead for Bill but not a landslide either. Debates go for Clinton but Bush holds his own but he just can't top Bill's Charisma. Bush's main attack point would be morality issues and it helps especially in the south but Bill campaigns on his record and what he wants to do with the deficit and foreign policy. The race is relatively close until Bush's DUI arrest comes out. This both hurts him with the religious right and the moderates. The DUI shows his wild side and makes him look like a hypocrite as he lied about it/refused to admit it and spent the whole time attacking Clinton. Clinton's campaign then goes on the attack late with statements and ad's which prove to hurt Dubya and he can't rebound from it.
In the end Clinton wins by a comfortable margin but Bush does better than his dad or Dole did. Nader takes the hard left away but doesn't win the youths as much as he did. Bush goes home to Texas defeated while Clinton gets another term. He feels a little better as the Democrats win the Senate and make gains in the house. But most importantly he gets a solid win on his own terms without the rat Perot interfering and a solid EV win with close to 350.
Clinton/Gephart-348
Bush/Cheney-190