I wonder what the goals of this fascist Turkey would look like? I could see some radicals call for the restoration of the borders of the Ottoman Empire pre-WW1 at the very least.
My take is a sort of dysfunctional republic of oligarchs is more likely. Assorted factions of the wealthy squabbling over this residual impoverished Turkish state. Some sort of facist party or doctrine may emerge, but I cant see this state as of any practical attraction to Germany or Italy in any major way.
I'm also looking at the potential for Greece in this. First off there is the cost of asorbing the Ionian refugees in the 1920s waived away. Second there is the economic benefit of owning Iornia. The net gain from these two is large in the context of the Greek economy in the 1920s. It makes Greece ultimately stronger, and may even waive away a Italian attack in 1940, thus extending Greek neutrality through WWII. If the Italians do attack their defeat may be such British intervention is unneeded & there is still a possibility of a neutral Greece during the war.
A third item affecting Greece is the status of Constantinople through to 1940 & beyond. It is not impossible Greece can gain control of that territory later in the 1920s or thirties.
Then there is the situation of the other territories sectioned off in the Severes treaty. Who retains control of those through the 1930s? In 1941 who controls the Chromium mines, & other critical mineral deposits in modern Turkey?