Greece decides to support Austria in the Third Balkan War against its fellow Orthodox Christian nations and allies; Serbia and Romania. As a reward, Greece is awarded Macedonia, formerly Serbian territory. This only wets the appetite of the Hellenic Kingdom.
The Armenian Genocide and Treaty of Servres
Though this TL does not truly have an "Armenian Genocide", it does have something similar. Appalled by Turkey's cruelty to their "Christian Brothers" (thought the Eastern Orthodox an Oriental Orthodox have had no love lost in the past), Russia starts another Russo-Turkish War. When the governments of London, Paris, and Rome hear word of this, they step in for a land-grabbing spree of the Ottoman Empire. A losing Ottoman Empire is forced into a treaty that greatly resembles the Treaty of Sèvres.
Constantinople was a problem, both Britain and Russia want to make sure the other didn't get it. Britain, proposed giving it to Greece. Russia, realizing it is unlikely they could actually keep the city anyway, agrees when they think that at least the Hagia Sophia is in the hands of a fellow Orthodox Christian nation. Greece is ceded the city only when it promises to remain neutral in issues between the two nations.
The Pogrom of Constantinople
The shining city of Constantinople was finally back in Greek hands. Within its walls the population was an equal mix of Greeks and Turks. That was subject to change. More radical groups began promulgating the expulsion and death to the Turks in Greek lands. So, on sunset July 13th, 1915 (the start of Ramadan), riots broke out across the city's districts. 300 Turkish citizens were killed. In an ironic twist, the city's Armenians and Jews actively participated, looting Turkish stores, and killing citizens. The Mayor of Constantinople himself even allowed for a demolition team to remove the minarets and begin taking off an Islamic artifacts in the Ayia Sofia, where they were sold for an ungodly amount. Though the LoN decried the situation and Ecumenical Patriarch Germanus V of Constantinople threatened any arrested with excommunication (dozens were disciplined, but as is the common practice after a 10 year process of penance some were readmitted). Within the week 60% of Constantinople's Turks fled to the rump state of the Republic of Turkey, occupying a landlocked area in the heart of Anatolia. A Greek census placed the Turkish population to 3000-4000, up 2000 from 2001. It should be noted the riots were not confined to Constantinople; Greek controlled Anatolia, Thrace, Epirus, Macedonia, and the Peloponnesos experienced similar attacks on Turks in the area, some riots even included the murder and looting of Albanians, Skopians (called "Fyromanians" in our world), Vlachs, and other minorities in Greece.
The Cyprus Issue
The main conflict in Cyprus is not between Greece, Britain, and Turkey. Turkey is much weaker and could do nothing to support the ethnic Turks of the island. In the mid 1900s, Russian actions in the Balkans is the boiling point for the Greeks to decide they need to avoid being in Russia's sphere of influence, and instead decide to strengthen relationships with Britain. Eventually, when the Statute of Westminster allows each nation to vote on whether to join the Federation or be completely independent. Cyprus chooses to merge with Greece instead of with the IFB. Enosis was complete on 16th August, 1960.