Very roughly the Middle Ages. We start getting references to Ashkenaz in the 10th century, and the first Yiddish we have is from the 11th century. Rashi, for instance, used a couple of Yiddish words, but when he wanted to provide local glosses for Hebrew/Aramaic terms, he mainly used French. A good primer might "Where Did Yiddish Come From?" and "The Mystery of the Origins of Yiddish May Never Be Solved". I haven't seen as good an analysis about Judeo-Spanish, but it's clear Judeo-Romance languages pre-date Judeo-Germanic ones.

So there start to be separate communities in the Middle Ages, but as far as I can tell, the differences do not become theologically decisive until Shulchan Aruch is published. Sephardim become the communities who accept the Shulchan Aruch without addition. Sepharad means Spain just as Ashkenaz means Germany, but one definition of Sephardi (such as Israel having a Chief Sephardic Rabbi and Chief Ashkenazi Rabbi) includes many communities without connection to Spain, like the Iraqi and Persian Jews. The Ashkenazi then become those who only accept the Shulchan Aruch with Rema's "mappah" (Shulchan Aruch means "the set table", mappah means "table cloth"). This is sort of the terminus post quem for the separation, which had been moving apart since the Middle Ages. Knowledge, however, continued to move back and forth, particularly in fields like Kabbalah, but Halacha become increasingly focused on two separate conversations.

More restrictively, Sephardim refers to those communities that adopted the Spanish and Portuguese Rite (sometimes called the "Western Sephardi Rite") and Mirzrahi refers to the rest of the Sephardim. These Sephardim in the restrictive sense obviously only go into diaspora in 1492, and are most common in Western North Africa (Morocco, Algeria, Tunisia), the Netherlands, the New World, and above all, parts of the Ottoman Empire (particularly Anatolia and the Balkans). Some communities, particularly those in Greece and Italy, have a separate history from all of them above. Greek-speaking Romaniote Jews in Greece kept a separate liturgical and halachic tradition from the more numerous Judeo-Spanish speaking Sephardi Jews.

In short, the legal, liturgical, and cultural divisions sort of overlap, but not completely, and we see different differentiations at different times.