Possible electoral shifts as a result...
2001: Mark Warner and Jim McGreevey are elected Governors of their respective states, as IOTL. Notably, however, Mark Green defeats Michael Bloomberg for the Mayoralty of New York City, absent concerns about national security. Giuliani's reputation, likewise, is not rehabilitated.
2002: As I've noted before, the bad economy is probably bad enough to put the Democrats back in control of the House and Senate.
Senate: Max Cleland (D-GA), Jean Carnahan (D-MO) keep their seats. Paul Wellstone retires, as he promised, after two terms, with no Iraq War looming in the background. Jeanne Shaheen (D-NH) probably beats Sununu six years earlier than she did IOTL. There's also the possibility that Wayne Allard loses in Colorado to Tom Strickland, and Erskine Bowles beats Liddy Dole in North Carolina. That puts the Democrats with a 54 seat majority, including, of course, Jim Jeffords. It's not stellar, but it's certainly an improvement.
House: Democrats can pick up as much as eleven seats here, putting the Republicans into the minority. Pelosi becomes Speaker four years earlier than IOTL. Hastert probably doesn't last long as Minority Leader, and is likely replaced by someone like John Boehner.
2004: George W. Bush beaten by Democratic candidate, but [insert candidate here] also has significant coattails.
Senate: Democratic pickups in Alaska (Tony Knowles), Illinois (Barack Obama), Colorado (Ken Salazar), Kentucky (Dan Mongiardo), while holding seats elsehwhere. Probably a 59-41 split similar to 2008's election.
House: Democrats probably increase their majority from 223 to somewhere between 230 and 245. Nothing earth-shattering.
Further than that, I'm not exactly sure. That all depends on how the new Democratic administration operates.