If I understand the OP correctly, the initial conditions involve a mostly Hellenophone Sicily and Southern Italy (how much there though? just Apulia, Basilicata and Calabria, or more?) in the early 19th century, while the Balkans and Anatolia remain under the control of a Turkish Muslim Empire (presumably still centered at Constantinople), against which local Greeks are revolting.
However, some other parameters need to be set in order to figure out what is going on:
1. What is the nature of the state which contains these Greeks? Are we talking about Latin lords ruling over a Greek countryside? Or a state where Greeks are actually politically dominant?
2. Extent and demographics of said state. It is safe to say it will have a sizable Latin minority sans very thorough Hellenization---Im not sure if that is even possible. Apulia, Calabria and Basilicata is probably easily done, but anything else will be a tough lift.
3. Circumstances behind the political divorce from Constantinople, assuming this is not alt Two Sicilies with a huge Greek peasant population (in which case very little should change by the time of the Greek revolution). Are we (under the butterfly genocide assumption) having a situation where Constans succeeds? Maniakes wins? Manuel does not fuck up? Norman conquerers of Byzantium get culturally conquered? These details will kinda matter with regards to understanding this state.
4. This is tied to above-but religion? I assume that they'd be mostly in communion with Rome, presumably under some Greek Catholic style compromise. Otherwise I am having a hard time seeing their continuing existence as a major chunk of the population. Rome and Constantinople were headed in different directions for quite a while pre Great Schism, so I think it is likely that something like the Great Schism will be inevitable, unless the Roman Empire has the means to politically pressurize the papacy.
If we go with my reading of the OP-we have a Greek Kingdom of Two Sicilies (with possibly a sizeable Latin minority) that is in communion with Rome. The origins of this are unclear, so let's just go with Basil II reconquering Sicily and stabilizing the area, and some sort of a 1204 style fracture of the Empire (Mongols take over Constantinople? Latins have easier prey in Italy than the Theodosian walls...) which leads to the Italian Catapenate becoming it's own state, under it's own Emperor (perhaps a Despot, but I doubt it---this will be the biggest of the fragments to emerge in all likelihood). History proceeds similarly (perhaps with this state taking over some of Venice's role in the E. Med). It will be difficult to knock out this Empire from all of mainland Greece (Venice put up a good fight OTL, and this one actually has the demographic depth to give the alt-Ottomans more bloody noses), but let's assume poor leadership at critical moments does the deal.
Even if the religious differences are huge, it is difficult to see a scenario under which Italian Greeks do not become a major part of the revolt, just due to demographics. Modern day Greece has 11 million people, Sicily alone has 5 million. Apulia (4), Calabria (2) and Basilicata (0.5) alone gives it 11.5 million in modern day terms-to say nothing of the rest of the "Empire" like Naples. Assuming similar 19th century population growth rates, which is very crude, but probably not too bad-they probably have more Greeks in there than the whole of Balkans, so it is an enormous pool of resources. The fear of the British (wishing to avoid an Ottoman collapse) or opposition from a large Latin minority may prevent the state from directly intervening (frankly, I doubt it-there'd be a new dynasty by the next week in all likelihood). There would however be large amounts of volunteers and material supplied accross the Adriatic/Ionian, while you'd have intellectuals from Italy kicking up a frenzy about Turkish atrocities in western capitals. It would probably be a huge boost overall, though some Great Power intervention would likely still be necessary-either directly, or just letting the Italians fight the Turks. The last one can end badly (since we know nothing about the ability of this state), but it is not unlikely that a Greece of not dissimilar size as OTL is formed in the end. Depending on the religious situation, it might be directly annexed by the Italian "Empire" (I can see the Brits and French prefering this-better the devil they know, plus fear of Orthodox Greece becoming Russia's puppet), turned into a Despotate/Kingdom under a member of the Italian dynasty or something more like the OTL Kingdom. However, I think enosis is very likely if the religious issue is not greatly played up---and the lack of trauma from alt Fourth Crusade (can't see that coexisting with Hellenophone southern Italy) might mean this is not as fertile a ground, especially when so many of their fellow countrymen remain under the Ottoman yoke. There would be a moment for it (probably whenever Russia overplays her hand like in an alt Crimean war), and it is also possible that they come to an arrangement with Austria over the corpse of the sick man in the Balkans. Aftterwards, hard to be certain. They'll probably go for Cyrenaica, and the Dodecanese easily, and might even have quick luck with Crete/Cyprus. The OTL economic hardships of the Kingdom of Greece may also be somewhat alleviated via Italian subsidies, which could mean better circumstances there. However, the Italian Emperor will almost certainly crave Constantinople, risking probably avoidable wars. It might lead to overall weird maps: Western Thrace and some of Macedonia being Turkish, while a Greek Smyrnan exclave holds on etc. Unclear how it ends, but this Greek state will be a much more formidable adversary than the Ottomans (of course, that might make everyone else dogpile on them instead, like Bulgaria in OTL 2nd Balkan war).
I think that the only thing that appears clear is that the ghost of Byzantium will not be put to rest as easily. The political continuation of the Greek Empire in Italy would mean that it would be really hard to try to go back to exorcise the Byzantine era from national consciousness (I know it was hardly successful OTL, but it will be a doomed cause here). Unless of course there exists a very deep religious gap between the states (Serb-Bosniak or Indo-Pak level) or the Balkan/Anatolian Greeks feel deeply betrayed by the Italians to have it in their national consciousness. Then we may well have "Greece" in the Balkans, while "Romans" continue on in Italy. Delicious irony, that one would be.
