Hillary Clinton would win the nomination.
If Edwards won Iowa, he'd gain a boost but given how weak his initial support was and how close the Obama vs Clinton race was IOTL, plus the affair, his chances are very slim. Obama and Clinton compete for New Hampshire, which results in a Clinton win there and then another Clinton win in Nevada. Edwards is struggling now having come third twice and needs a South Carolina win to upend the race. But Clinton amasses a coalition of South Carolina whites and blacks to eek out a narrow victory over Obama and Edwards. Heading into Super Tuesday, despite her Iowa defeat, Clinton is the clear frontrunner after her New Hampshire comeback. Then in the final nail of Edwards' coffin the affair is leaked and he does very poorly on Super Tuesday. Obama does better, winning many caucuses but he is still behind Clinton. Clinton wins many black voters, who are convinced that Obama isn't able to win white voters. Hillary has a delegate lead of hundreds of delegates and wraps up the nomination.