POD: Bob LaFollette doesn't get sick and make a rather long, rambling, and somewhat insulting speech to the press corp in 1911. He enters into the race stronger, and is able to make peace with Roosevelt (TR: "For the last time, I am sorry about spilling that tea on your wife." LaFollette: "We'll deal with that later.") The combo of LaFollette and TR overcomes Taft at the Republican convention, and Robert Marion La Follette is the Republican candidate in 1912.
La Follette defeats Woodrow Wilson is a, not really that close election, and becomes President. As part of the peace accord between him and Roosevelt, he choosed Hiriam Johnson as his VP and agrees to nominate General Wood as his Secretary of War.
Much to TR's disgust, La Follette isn't the type to be cowed. A boxer in his own right, and someone who finds himself just as manly as the former president ("thank you, very much") he refuses to brown nose after the election is done.
Horrified by the outbreak of war in 1914, La Follette pushes for a series of Neutrality Acts; think those from the 1930s, if not a bit stronger. He only prosecutes companies which are giving loans to either Great Britain or Germany.
However, La Follette isn't really 'anti-war' so much as he is, anti-THAT war. He finds himself drawn into the Mexico mess on the side of the Republican. The US ends up victories, helping set up a strong Republic in its southern neighbor, but only after several years of fighting.
He wins reelection in 1916, but sees a Democratic landslide in 1920. To this day, the American people are split over his legacy, with some claiming him to be one of the greatest Presidents of the 20th century, and others bemoaning his election.