I don't know much about the Serbian or Russian cases, but the Bulgarian case was explicitly linked to nationalism. The Greek Orthodox Patriarch of Constantinople was an official of the Ottoman state and responsible for administering the affairs of Ottoman Christians. This office was controlled by the Phanariots in the 19th century. This was a group of Orthodox based out of the Fener district of Istanbul. You could work your way up in this system if you were Orthodox Christian and learned Greek.

Of course a great many Orthodox in the Ottoman Empire spoke Greek as their mother tongue. But many were native Bulgarian speakers. For nationalist reasons, int the late 19th century, Bulgarian speakers demanded a separate (autocephalus) church that would specifically administer Bulgarian Orthodox Christian affairs. A major reason was because Greek nationalists claimed anyone under the Orthodox church in the Balkans should be considered Greeks, regardless of language. This would give them a claim to a large portion of the Balkans. The Ottomans acceded to Bulgarian demands and created the Exarchate, which was the equivalent of the Greek Orthodox Patriarchate, but for Bulgarians. They built a cathedral on the Golden Horn and everything. Once the Bulgarians gained independence, they took their church with them.

I would suspect the Russian split came when the Ottomans conquered Constantinople and made the Patriarch of Constantinople an office in the Ottoman government. Hopefully someone with expertise on that era will show up and comment!