We'd need to go back to Nader's heyday, the aftermath of the publication of Unsafe at Any Speed, to give him a shot at this. Also, we need a few small points of diversion rather than just one, and it takes a while:
Riding a wave of popularity in liberal circles, Ralph Nader decides to come back home to Connecticut in 1969 in order to run for the Senate in 1970. Entering an already crowded primary, Nader manages to make a splash, convincing Joe Duffey, a popular liberal Democrat who was considering running, to decide against it. Tom Dodd, the Democratic incumbent, dies from a heart attack in June 1970. (In OTL, Nader continued living in DC, Duffey got the Democratic nomination, and Dodd dropped out of the Democratic primary race after a non-fatal heart attack, then ran as an independent.) Nader was appointed to the now-vacant Senate seat by Governor John Dempsey, and he went on to win the Democratic nomination. Thanks in large part to nationwide support of Nader from various prominent liberal Democrats, the advantage of his new status as the incumbent, and failure by the Republican candidate, Lowell Weicker, to win over the more moderate former supporters of the deceased Dodd, Nader won a close election that November.
By 1976, Nader was 65th in seniority in the Senate. That year, he ran against former Lt. Gov. Peter Cashman, who Nader defeated handily. In 1984, Nader was supposedly being considered for the Vice Presidential nomination by Walter Mondale, but Mondale eventually decided against him. In 1988, Nader was selected by Joe Biden as his running mate in the election against Republican nominee Bob Dole. Biden and Nader narrowly won the election. However, throughout his Vice Presidency, Nader drew the ire of the increasingly conservative Democratic Party. In 1992, Biden and Nader won re-election very narrowly. Vice President Nader teamed up with First Lady of Arkansas Hillary Clinton in an effort to enact Health Care Reform. The plan was a miserable failure, but endeared Nader and Ms. Clinton to the left. The 1994 "Republican revolution" marked an apparent end to the left wing, and in 1996, Bob Dole won the Republican nomination yet again, roundly defeating Ms. Clinton's husband, Gov. Bill Clinton, in that year's Presidential election.
After his Vice Presidency, Nader publicly renounced the Democratic Party. In 1998, he joined the Green Party, and encouraged the nation's liberals and progressives to do the same. In 1999, Nader announced he would run for the Green Party's nomination for President, which he would go on to win. He selected Dianne Feinstein as his running mate. In 2000, President Dole decided against running for a second term, citing his age. The Republican Party nominated former Secretary of Defense Dick Cheney, who chose Governor George W. Bush as his running mate (admit it, it's far more plausible than what happened OTL). The Democratic Party nominated Former Gov. Evan Bayh, who chose Senator Joe Lieberman as his running mate. The moderate ticket floundered between the ideologically charged Cheney and Nader, and Bayh, while young and energetic, had a difficult time establishing a coherent base. In the debates, Nader moderately outperformed the other two, while Feinstein completely destroyed Bush and Lieberman. Marginalized by the strength of the Green ticket, Bayh realized that he would simply take votes from Nader, and began to campaign only in states Cheney was likely to win.
On election night, Nader and Feinstein won a clear plurality of the popular vote, 47%. Cheney and Bush won 32%, and Bayh and Lieberman won 21%. Nader won a slim majority of the electoral college, earning the Green Party the White House for the first time. President Nader served a single term, from 2001 to 2005. He was noted for his introduction of universal health care and his expansion of welfare coverage. His cuts in the defense budget allowed him to do this while actually shrinking the national debt. However, it also opened him up to an extreme degree of criticism after a terror attack on Washington, DC and Los Angeles on March 11, 2004, destroyed the Pentagon and the Library Tower and claimed thousands of lives. Emboldened by the calls for a larger defense department, John McCain won the Republican nomination, going on to win the Presidency against Feinstein and Democratic Nominee Al Gore in 2004.