Only way I could see this happen with OTL Nader is if he does better in 2000 and Gore somehow manages to win. If the Republican wins, Nader will be blamed, regardless of how accurate that charge really is.
Gore wins, and dies on the equivalent of 9/11. President Lieberman, expending his vast political capital, invades Afghanistan and Iraq. Republicans are critical of the conduct of the wars but are by and large supportive of them. The Republican anti war faction is slightly bigger here, but the thinking of both parties is largely supportive of decisive action being taken against America's enemies.
Lieberman is able to fend off a surprisingly strong anti war challenge from Ohio rep. Dennis Kucinich and wins the 2004 nomination. Kucinich refuses to endorse Lieberman, and he and his supporters leave the party. He is reelected to his seat as a Green, becoming the first third party candidate elected to congress in some time. Lieberman goes on to defeat the Republican and Nader, who stays with the Greens given his better success in 2000. The Greens actually crack 5%, gaining them federal funding for the 2008 election. Kucinich remains their sole representative, but several Democrats in liberal areas are defeated in primaries for their support of the wars. The new Democrats refuse to support Lieberman in just about anything he does.
2006 is a bad year for Democrats, and Republicans pick up many seats. Lieberman's' approval ratings sink drastically, and when he invades Iran, protests not seen since the Vietnam war arise in many cities. In 2008, Democrats nominate Hillary Clinton and General Wesley Clark. Republicans nominate Senator Rick Santorum and John Bolton(or someone equally hawkish)With the Republicans strongly in support of finishing the job and Democrats lukewarm and not giving an exit date for troops to come home, The Greens see an opening. In a huge convention in Madison Square Garden, they nominate Ralph Nader, still very much the face of the party, and Rep. Kucinich.
It's a tough three way race, but the chain of president Lieberman drags the Democrats down, and Ralph Nader is narrowly elected the 45th president.