Here are some "really late" PODs, not sure how plausible they are...
It is around 1820, and all of the major powers-that-be (UK, France, Russia) decide that the Ottomans are to be dismantled to good. The Greek independence is a good enough rallying cry, and when the British warships start pounding Constantinople with their cannons, Greek army (much better organized than in OTL) approaches from the South, French one from the West, and Russian from the north, the Sultan has no choice but to surrender completely, whereas he is transported to some hellhole in the depths of Siberia where he lives off the rest of his life in relative obscurity.
The Ottoman territory is then divided between the powers; nominally it is considered reinstituted Greek Empire (or neo-Byzantium, should one call it that); however, it is considered a joint protectorate of the Three Powers, with Constantinople itself a neutral city, and various zones garrisoned by troops of different countries - France in the Balkans, Britain in Asia Minor, Russia in Northern and Eastern Anatolia and Caucasus. Egypt is given to Britain, while some other North African conquests are given to France. Russia gets Caucasian territories.
Italy gets reunited on schedule, similar to OTL; WWI starts similarly enough (although the names might be a bit different, and the cast of characters would vary). Italy stays out, whereas the "Greek Empire" (in which by now the Greeks assume more and more actual, and not figurehead power) is fighting Austro-Hungary with minimal effort. The Russian Revolution occurs pretty much on schedule, however, the Romanovs are able to flee to Constantinople, where, after much lawyering, they are able to assume the throne after death of Greek monarch without any heirs (Romanovs being the descendants of at least a few of the Byzantine Emperors - Constantine IX in particular, and probably a few more).
Italy joins on the Entente side when their victory is getting clearer, and end up getting a few concessions in the Balkans, which the Greeks are none too happy about. This results in "Greece" breaking all ties with remaining Entente powers, and becoming increasingly isolationistic, and relying on its own devices. The Russian civil war draws on, resulting in breaking off of several outlying regions, some of which claim allegiance to the Romanovs.
WWII still happens, although here, Italy decides to stay neutral, and so does "Greece". WWII is longer, bloodier, and exhausts all sides involved, with the issues not being resolved by the end - pretty much think the ending of WWII is similar to the ending of OTL WWI. In a meantime, Italians and Greeks alike enjoy higher living conditions, and make a monetary killing through selling weapons, supplies, and whatnot to both sides. The United States' interference is balanced by Italy and Greece being rather opportunistic, and expanding their own spheres of influence, expanding into the Middle East in the latter's case, and into North Africa in the former's.
As the Cold War draws on in Europe between the Fascist bloc, the Communist bloc, and the Capitalist bloc, Italy and Greece remember old animosities, and enter into a cold war of their own, thus building up their powers and both acquiring nuclear weapons some time in 1960s. As the Fascist and Communist blocs collapse in 1990s, Italy and Greece pick their respective pieces, and emerge as rival superpowers, and rival to the not-as-dominant United States.
