Nearly the same as OTL, with one important caveat of there being no war scare. Napoleon, unlike Britain, isn't in a position to deliver any kind of ultimatum, and most likely puts considerable diplomatic pressure on Washington (with the support of London, Berlin, Vienna, and St. Petersburg) that it be done.
Lincoln delays until the public is prepared to accept the release of the prisoners, then they are promptly put on another ship to the Continent.
The only post event difference I could see is that France, with her dander already up about the blockade and the resulting cotton famine, probably leans distinctly more pro-Confederate in the aftermath. Plenty more looking the other way as weapons and material are smuggled to the South, hell maybe the CSA gets a few shiny new ironclads to play with.
In the long term it really doesn't change anything about the war, but in the short term the Union has a bit more of an uphill battle.