Oldest mention of Croats is from mid 9th century, the charter of duke Trpimir who calls himself duke of Croats, and where he acknowledges the land grants of his predecessor Mislav.

Oldest mention of Serbs is in Royal Frankish Annals from 822 which mentions that Slavonian (modern north Croatia) duke Ljudevit, who rebelled against Franks, fled to Serbs.

Byzantine state manual De Administrando Imperio written around 950 describes Croats and Serbs and mentions how they both came separately and independently to the Balkans from White Croatia and White Serbia which were neighbors somewhere beyond the Carpathians.

While accounts in DAI are of doubtful authencity because they describe second hand accounts taken from Croatian and Serbian oral history, the point of it all is that Croats and Serbs exist as separate identities for at least 1200 years and possibly a few centuries longer.

But then we come to a problem with your question. Did Croats and Serbs split from some common identity, if yes when and where?

We don't really know if ancient Slavs thought of themselves as one common group, or they thought of themselves primarily by tribal distinctions. Or both? Maybe different groups at different places had different feelings about this? Who knows.

The question of other Yugoslav ethnicities is more complex and controversial, but a matter of fact is that there wasn't one Yugoslav nation which split up. The matter is far more complex than that.