Moving back to the TL & the Med, I don't see a viable alternative to the Allies invading Italy although I agree with Usertron2020 that it goes nowhere strategically after you reach the Alps. The problem is that the same applies to Southern France (1) & as for a Dodecanese campaign... Ugh! Hopefully Alan Brooke would have shot that idea down if Winston raised it again; didn't that man learn anything from Gallipoli? (2) The geography of the area sucks & any campaign there will just swallow manpower into a meat-grinder that the Axis can reinforce faster... a landing from the Black Sea (somewhere between Varna & Constanta) then up then up the Danube Valley has some promise; the drawback is it's a rather elongated supply line even IF the Turks will allow the Allies passage; I rather think not IMO. (3)
Taking Sicily, Sardinia & Corsica has promise... it pretty much opens the entire western Italian coast for landing & puts the entire peninsula in range of Allied aircraft (4) (if they can stop the Italians bombing it!).(5)
1) Not so much as the Aegean. If done well after Sicily, Sardinia, and Corsica are taken, secured, and built up, that is. Southern France presents problems, but a good breakout from the landing zones, with Marseilles and Toulon taken and fully operational (no serious Atlantic Wall down there)? It promises to draw so much weight from the Germans that even the Stalingrad Campaign could be seriously affected, at the very least in terms of trying to relieve the German Sixth Army.
2) No, he didn't. His blindness to operational realities was absolute. Part of his problem with Australians stemmed from his blaming them for Gallipoli's failure, when they were in fact the least culpable of all. I would suggest that even today the defenses of the Dardenelles
of 1915 would provide a very serious deterrent to an invader coming come the Mediterranean Sea. For logistical reasons if no other-no ports in the straits this side of
ISTANBUL!


And the nearest other port (though a large one) is on the central Ionian coastline (Izmir).
3) IMVHO, you're right. The Turks were ferocious in their independence and neutrality. They considered keeping the Bosphosrus closed to wartime naval belligerents as a statement of their own national sovereignty. They played along with the Germans, but made it quite clear to Hitler they weren't joing the Axis this side of a 1940 British or 1941 Soviet surrender. They also made it clear later in the war that they were going to have to DoW the Axis just to survive. Which they did, on February 23rd, 1945.

What sloth!
4) A very Pacific strategy for the Royal Navy. Which TTL makes quite workable.
5) Uh, I wouldn't worry about THAT particular part of the Axis air equation. By the time the British could finish such a campaign, there would not be much left of the Regia Aeronautica.