It could be as late as c. 1900. Have the Ottomans hold onto some more of their Balkan territory, in addition to Anatolia, Mesopotamia, the Levant and the Hedjaz. Have them sit out the First World War, if such occurs, or at least have them on the winning side.
Then just have them wait until oil. Now we have a G8 level nation that has control over significant reserves of THE strategic resource. If a Cold War between Russia and America happens, they'll be on the side of the United States against their old enemy, as a Japan, Britain or France caliber ally.
I don't know what the PoD would be. But the collapse of the Ottomans, or the Austro-Hungarians for that matter, was not inevitable.