I suspect that the Ottoman Empire would have slowly fragmented into regional groupings of Turkish, Arabic and Farsi
speaking areas at a later stage, but without active western intervention that might have taken a long time.
A key question is how the British, French, Italian states would make arrangements to obtain oil.
Persia was never part of the Ottoman Empire. The british got most of their oil from it out of Abadan.
The french might have put more energy into finding oil in Algeria, or they might have bought it from a turkish Iraq. Not much change I think. Assuming France and the UK won WW1 then the ottomans would not be allied with defeated Germany anyway and french money would be welcome, as were french products.
Italy... eh, who cares? They'd buy it from someone, same as in our timeline.
@r16, in the very unlikely case of the Ottoman Empire remaining neutral, I believe the french and british appetites for colonies would probably be quite quenched by the end of the war. Just as it was in out timeline, otherwise Turkey would not exist today, they'd invade and dismantle it if the war weariness among the population and war debt weight on their treasuries was not so high. But it was hence no hurry for new colonial acquisitions, especially acquisitions that looked like more trouble than profit.
Churchill basically handed over the Hejaz to the saudi desert barbarians rather than spend some more pounds to defend it. The english had the Trucial States ad Kuwait, plus Adem, that provided control over regional trade. No need to expand further. They held the most profitable bits which were cheap to maintain, without having to spend on repressing masses of rebellious populations across another wide area. Sure the british kept Iraq and Palestine but that was them having to have something to show for the war effort. Palestine was effectively handed over to jewish settlement also, and Iraq puppetized.
The french did seem to have an appetite to cling to Syria and Lebannon, crusader urges in some late romantic revival? But they were the first to back off from carving up the rump ottomans and even handed back a province. No appetite for more war either.
So the Ottoman Empire would probably endure between the wars. And even get propped up by the post-ww2 europeans (assuming a ww2 happened and ended as it was...) because they'd be a non-european empire showing those amerians and soviets that colonial empires were not just an european obsolete idea. Extremely useful fig leaf? That Ottoman Empire, surely reformed into some constitutional thing, could play the role of a huge Israel... it would mess with the american UN idea and gobbling up of the Middle East, that is when this alternate history would get really interesting...