Here's where things get
very interesting. Strictly speaking, in Greece there's not just 1 Orthodox Church but at least 3 (if one is excluding Mount Athos):
>the Church of Greece proper, whose canonical jurisdiction is Greek territory prior to the Balkan Wars (aka "Old Greece");
>the Ecumenical Patriarchate of Constantinople, which has jurisdiction throughout Northern Greece and the Aegean islands (the "New Lands", which are managed by the Church of Greece via an agreement), including the Dodecanese (where the Patriarchate exercises direct jurisdiction, due to that island group's separate history as an Italian colony for part of the 20th century);
>and the
Church of Crete, which is its own thing altogether.
Now, if it were up to me doing it (and boy have I had an eye for this, so
gardez-vous!), everything all depends on three factors: α) how much territory the
Provisional Democratic Government and its military arm, the Democratic Army of Greece, controls; β) the ideological orientation of the KKE (as it was influenced by both Stalinist and Yugoslav currents - the latter being important for obvious reasons, considering the Balkans); and γ) how each of the Orthodox Churches react and thus what the relations are between the Provisional Democratic Government and the Greek Orthodox Church in general. For the problem here is that during this period even the most atheist of Greeks is also a committed and devoted Orthodox Christian as the Greek Orthodox Church is woven into modern Greek identity to a degree not understood from a North American or Western European POV. Even Roman Catholics and Eastern-Rite Catholics (both exist to varying degrees in Greece) are also culturally Greek Orthodox as that is part of the culture. Now, it is generally true that throughout Eastern Europe the Communist governments usually treated their Eastern Orthodox Church like shit, particularly by co-opting them and infiltrating them with members of the secret police and thus turning the Church into a control mechanism for the vast majority of faithful. In Greece, that's not going to work too terribly well, for obvious reasons - while infiltrating the Orthodox Church could work for the Church of Greece, it's
not going to work for the areas controlled by the Ecumenical Patriarchate, nor for that matter the Church of Crete.
For this exercise, I'm going to assume that the Provisional Democratic Government manages to control the whole territory of Greece (including the Dodecanese and Crete) and the KKE readily embraces both its Titoist and Stalinist currents, thus making Greece neutral when it comes to the Tito-Stalin split and thus can play both Yugoslavia and the Soviet Union off to Greece's benefit. As a cradle of Greek identity throughout the ages, the KKE would naturally want the Greek Orthodox Church as its ally (after all, even Stalin rehabilitated the Russian Orthodox Church during WWII). So Communist Greece could utilize what I'm going to call "pragmatic atheism". That is, although atheism would be state policy, in reality it will require so many qualifications that it would basically become de facto agnosticism. In practical terms, that would essentially mean a Greek version of laïcité that would comply with Marxist teachings as well as cordial relations with the Greek Orthodox Church (one does not necessarily preclude the other, as its Aegean neighbor Turkey demonstrates). To this end, all the pre-existing arrangements between the Church of Greece and the Ecumenical Patriarchate can continue, with only minimal interference by the state, as would the continued existence of the Church of Crete. There would obviously be some politics involved (as there always is when you have a state church like the Greek Orthodox Church), and as I stated earlier the Church of Greece would definitely be infiltrated (but not the Ecumencial Patriarchate's territories), but for the most part it would probably be a better situation than most Eastern Orthodox Churches in Communist territory. Where things would get tricky would be Mount Athos, which would require a lot more explanation than space would dictate, but the Holy Mountain is basically an independent republic under the direct jurisdiction of the Ecumenical Patriarchate, governed by its monasteries, with only basic representation by the Greek government as it borders Greek territory and all monks living on Mount Athos are either Greek citizens or are granted Greek citizenship upon arrival. That would have to require some delicate negotiation between Communist Greece and the Ecumenical Patriarchate.
That's the TL;DR version, positing a best-case scenario. What would actually happen would require a much longer post going into all sorts of detail that it would bore everyone to death, but even then I would still assume that the Greek Orthodox Church would still be much better off than elsewhere in Eastern Europe, although it would be much more complex and complicated.