George McGovern won the nomination by a frog's hair. He narrowly won the California winner-take-all primary against Humphrey. Winner-take-all primaries were against McGovern's own new rules for the DNC, but he ironically and successfully shut down the Anyone But McGovern attempt to proportionally distribute the California delegation at the convention. The only primaries after McGovern-Fraser that were anywhere as close as the 1972 Democratic primaries were the Republicans in 1976 and the Democrats in 2008.
Humphrey is the most likely nominee. Humphrey won the popular vote and tried to win at the convention, so he was basically the Hillary Clinton analogue (for the closeness of the race in the primaries' popular vote)
and the Ronald Reagan (for the closeness of the race at the convention). Had he been nominated, I would consider the race a coin flip. I'm surprised there aren't a lot more Humphrey '72 timelines out there. If the ticket is Humphrey/Wallace (which is very likely, see below), Nixon is in
serious trouble.
Had George Wallace not been shot, he would have tried to become king of the convention, but most likely would have ended up kingmaker. He probably ends up winning the popular vote in the primaries, and Humphrey/Wallace ticket that was vaguely considered IOTL probably becomes a reality. Humphrey combining with Wallace = game over for McGovern. You could get him to win the nomination with some PODs before his shooting though; of all the candidates in 1972, he was the most powerful. What would later be known as the Reagan Democrat was back then called "the Wallace factor," and he came second crucial Wisconsin primary, which McGovern invested pretty much his whole campaign in over a year in advance. Wallace only campaigned in Wisconsin for one week, halfheartedly.
Had Wallace gotten McGovern's organization, he could've potentially defeated Richard Nixon by a nice margin. A Wallace nomination was Nixon's greatest fear for this very reason. If Wallace fails to get enough of his way at the convention, he could've started another third-party campaign. IOTL he didn't because a McGovern nomination plus the Eagleton disaster made the outcome of the election inevitable, but I could see Wallace bolting if Humphrey won the nomination easy enough to ignore Wallace's input.
Muskie is a paper tiger: imagine Mitt Romney in a field of heavyweights.
Of course, you could also get John J. McKeithen.