Even if the Turks lose their war, I'm not sure the exact map above
can become a reality. The Entente powers aren't up for infinite application of military force (as their OTL stances after World War I show us in some detail), so even if the Turks gets crushed, the main things you'll see is that authority in the West gets handed off to the Greeks, and in the East to the Armenians.
Regarding the Greeks: they fought hardest in OTL, as you've noted, so it'll mostly be a case where they declare the Megali Idea realised and wield effective power over the Straits, and the Entente powers ultimately just recognise this is a fait accompli (after the Greeks ensure free Entente shipping across the Bosphoros, and agree to participate in the Western blockade of the USSR).
The Greeks will simply receive a lot of Western backing, and so will the Armenians. To the effect that Armenia ultimately looks like this:
(Excepting, I expect, the yellow area North of the Lori region. That'll be Georgian/Soviet occupied soon enough.)
The rest of the ideas is considerably more questionable. I don't think the French will be willing or able to project power deep into Anatolia. The Italians will be willing, but not able. I expect that their zones of control will be reduced to coastal strips-- but in the process (and also in the context of them winning the 1920 war against the Turks), their control
will become more entrenched and consolidated.
A plausible scenario is that in the West, the Greeks extend their holdings to Marmaris and the isle of Rhodos, while in the East, the French just outright annex everything along the coast Westwards from their territory along the Gulf of Alexandretta, up to Silifke. The Italians fully occupy the Southern Anatolian coast in between these two points, with Antalya becoming their colonial headquarters.
The Kurdish region most probably gets its idependence, but the Anglo-French control over the region soon becomes nominal, and the region becomes the source of cross-border raids into Syria and Iraq, and the centre of broader Kurdish independence struggles. (The USSR may soon seek to support the Kurds in these efforts.)
Now, assuming that anything like the above is realised, we end up with Turkey and Kurdistan both gravitating towards the USSR, since both are revanchist against the Western powers and against the Armenians. (Also, consider that the Kurds were, in practice, the main perpetrators of the Armenian genocide. Serving under Ottoman command, yes, and identified as "mountain Turks" by the Ottomans, but they were the ones who did a lot of the killing. Which makes sense, because a lot of the area was ethnically mixed between Kurds and Armenians.)
Meanwhile, Greece and Armenia are doggedly pro-Western, and solidly Western-backed. Italy likewise maintains a position that's more in line with the Western interests. And being occupied, uh... occupying the Southern coastal strip of Anatolia, there's a chance they have no means available for any adventuring in Ethiopia. So there's a half-way decent chance of Italy remaining Western-aligned.
There is even the potential that a Western-aligned Italy goes all-out on guaranteeing the security of Austria, meaning the Anschluß is ruled out. Assuming Italian fascim and Austro-fascism (or cognates thereof) still arise, which I consider probable, a long-term implication may well be that
national socialism ends up viewed as uniquely evil, whereas
fascism is considered far more normal and acceptable.
In these circumstances, the Nazi-Soviet pact may even last (at least last
longer), as both prioritise the matter of fighting the "decadent, perverse West" first and foremost. Which may in turn flip matters in East Asia, with China joining this "Continental Axis", while Japan instinctively sides against Russia.
(Mackinder be like: "
Told you so!")
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Apologies in advance for this off-topic line of discussion. I couldn't help myself.
Quick aside: Germany only attacked Yugoslavia because Yugoslavia refused to allow them to march through on their way to Greece (to bail out Benito). No Italian war against Greece = no need for any of this = no German invasion of Yugoslavia, either.
Persuant to the above, the Germans could
theoretically launch Barbarossa six weeks earlier, with at least two more tank divisions (I think four are freed up, in fact, although logistics may get in the way of using them all in the initial stage of Barbarossa)
and additional 20.000+ men that were killed in the Yugoslav and Greek campaigns.
Granted, those six weeks are theoretical, because the rains went on for pretty long that year. But they can use the four weeks to optimise their positions and preparations.
...Not that it makes even the slightest practical difference, because Hitler was a fool who would still dither about the choice between driving for Moscow, or doing a two-prong movement towards Leningrad and Stalingrad. He wasted two weeks on that (after Barbarossa had launched), basically cut his army in three, and only then had the centre drive for Moscow.
Whose outskirts they reached the very day that it began to snow.
If Adolf the Foolish had avoided the Greek/Yugoslav boondoggle
and committed to "
everyone, aim for Moscow!" from the start, he could thus have had:
-- Four weeks extra to prepare the attack adequately.
-- Four weeks extra to march on Moscow.
-- Two-to-four extra tank divisions and upwards of 20.000 extra men for the invasion.
-- All his forces available to take Moscow, with only a minority of the OTL Northern and Southern forces available to guard his flanks, and the rest free to join in the spear head.
Which means he'd have taken Moscow, destroying all effective logistics in European Russia (since all their rails went to and from Moscow), and forcing Stalin to flee East in a panic. (Stalin, in OTL, was at the train station, ready to evacuate, when the German assault faltered.) In short: then the Germans would have effectively won in the East by the close of 1941, because Soviet leadership would be in disarray and their logistics would be shattered.
But to manage all that, Adolf the Foolish would have needed to be Adolf the Clever, which he wasn't.
(Like I said, this line of discussion is off-topic. To avoid derails, further discussion should presumably not occur in this thread.)