Here's the thing for Greece. Whoever becomes King must convert to the Greek Orthodox Church, no ifs and or buts. That does not necessarily square well with the UK, which at the time the throne was de jure banned for Catholics and de facto the public expects an Anglican on the throne since whoever occupies the monarchy is Supreme Governor of the Church of England. Now converting to Anglicanism would be not that much of an obstacle to climb for a Hapsburg (the 19th century saw the rise of the Tractarians and the modern Anglo-Catholic movement which ultimately colonized the "high church" wing of the C of E; at this stage, even considering the context of the time, any attempt at reconciliation between the two would be most welcome). It's Greece that would be more problematic, since at the time most Western Europeans had this very rosy-eyed view of the place based on
Ancient Greece, which is as different from Ottoman-era Greece as Victorian Britain is from Roman-era Britain. The paradigmatic example (apart from Lord Byron) of this clash of interpretations of Greece is
King Otto himself, who made himself very unpopular and was forced out by the Greeks themselves. Any Hapsburg monarch of Greece would have to be separate from the British connection, number one, so that he could convert to Greek Orthodoxy in a sincere manner, and number two would have to considerably lower his expectations of what to expect for Greece and use what was already there. One big butterfly here would be no autocephalous
Church of Greece, but that would be no big problem; your average Greek would not notice the difference when s/he piously walks into a church.