In place of the Babylonian Captivity and Anti-Popes, you have a urope struggling to recover from Mongol depravations, Black Deathand local warlords. I picture a Western and Cental Europe of one province nations with the notable exceptions of Tartar Hungary and Plantaganet England under the Black Prince ruling parts of Scotland and France.
The Catholic Church would still have Bishops etc..., but their flocks would be animal as well as human souls. Orders such as the Franciscans with their emphasis on practical Chrisitianity would profoundly affect all other orders.The Knightly Monastic Orders Templars (clearly not suppressed), Hospitallers, Teutonic and Livonians would maintain their estates and way of life better than most.