What would happen if two men went on trial for the same murders but they convicted the wrong man and the man that wasn't convicted later admitted it?

The man that wasn't convicted would remain free, as double jeopardy prevents the prosecution from bringing the charge against him again.

The man who was convicted would likely file a motion for new trial and ask the court to overturn the conviction based on the new evidence. Depending on the strength and credibility of the new confession, he may or may not be released. There is no automatic process, and if there is any evidence to suggest that the new confession is made up, the convicted man may not be released.

In reality, two defendants would not be accused and tried for murder unless there was a suggestion that they were acting in concert. Murder trials are expensive and lengthy, and prosecutors do not take them to trial unless they have relatively certain evidence to suggest they have the right guy. This is not to suggest that everyone on trial for murder is guilty, but more that in a situation where the evidence is ambiguous, the prosecution is going to do more research and choose one defendant, rather than split resources prosecuting two.