There were Edwards staffers who knew of the affair and who agreed among themselves to leak news of it to the press should Edwards appear to be a serious contender for the nomination. The feeling was that if they did not and Edwards became the nominee, the Republicans were likely to find out about it and derail his campaign. I believe this is discussed in the Halperin/Heilmann book "Game Change".
With this in mind, it is likely that an Edwards win in Iowa would have led in short order to the implosion of his campaign in scandal. As to who would have won the nomination in that case, it is arguable, and the analysis depends to some degree on which candidate finishes second in Iowa.
Under the premise that Clinton finshed third (as was the OTL case and the one put forth by the OP) you can make two arguments. On the one hand, you can make the argument that Obama needed a win in Iowa to be viable and his campaign was indeed based upon that premise. On the other, the Clinton campaign was a cesspool of intrigue and incompetence that failed to place anyone in a senior role who actually understood the mechanics of the nominating process. All things being equal, with Clinton finishing third, I would give the edge to Obama on the basis of a superior strategy for accumulating convention delegates, as the Obama campaign had been organizing in key caucus states for some time prior to Iowa. In addition, the Obama campaign was better financed and had better financial management. Clinton spent foolishly and extravagantly in Iowa and already begun to develop cash flow issues after that contest.
Under the premise that Obama finished third, which is not altogether unreasonable if Edwards won given that they were competing for many of the same voters, the outcome becomes less clear. Obama would have been badly hurt by a 3rd place finish; the wholesale swing of black voters to Obama did not really take hold in the polling until after Iowa. Even then, though, the flaws in the Clinton campaign are still there. Unless Clinton won decisively in the February 7 Super Tuesday contests, there is a good chance that one is still left with a long delegate slog with the outcome in doubt. I'd give the edge in this contest to Clinton, but it would be by no means guaranteed unless she came close to running the table on February 7.