Hejaz and Yemen were under the control of the Ottomans in WW1- and, revolted from the Ottoman Leadership. Unlike the the Arabs in the Fertile Cresent, who the British betrayed to expand both French and British holdings, the Hashemites remained in Hejaz.
The Hashemites also took power in Jordan and in Iraq, and in Hejaz were allowed to declare themselves King.
Though the territory was technically within British SOI,
The Arabian Peninsula was actually just straight up ignored; I assume because everyone regarded it as a massive blank space with no value in its partition.
I would assume the territory, if partitioned, would enter Italian control- if nothing else, to pacify them after the Treaty of Rome was broken (though the land was poor in resources and economic potential at the time.)
It wasn't going to Italy, because pretty much nothing was. France was set against giving Italy anything, Britain not keen on the idea either.
So why not, and what if it was?
Basically answered it above; the area was generally ignored.
No, they didn't. They just betrayed them when they partitioned the crescent.
The issue is a lot more complicated than this. If you look at what actually ended up happening, the Hashemites ended up with a lot of the territory they might have gained; both Jordan and Iraq fell under Hashemite rule. The latter is particularly interesting, because the first King was Feisal; the same man who had been promised Syria and had been kicked out by the French.
And the idea was more to pacify Italy.
Very few people were quaking in their boots over a nation which had, to the world's view, just spent three years in futile attempts to take out the laughing stock of Europe.
Yemen was more economically viable than Hedjaz- so at least they could give the Yemeni portion to Italy- being right off from Eritrea and all.
Much of Yemen was already under British influence. As for Italy, the same point is again raised; why?
There was a bit of naivety around certain British actors in the region at this time (Lawrence, Gertrude Bell, and I believe Arnold Wilson as well) that the Arabs would not only meekly submit, but actually be pleased to accept a British paternalistic hold over their affairs.
Lawrence was actively in favour of the Arabs, and tried his best to get Feisal installed as King of 'Greater' Syria (roughly comprising Palestine, Syria, Lebanon, Jordan, and Northern Iraq).
Wilson was a paternalist through and through and felt the Arabs should just be ruled directly as any other colony, thereby experiencing the wonders of enlightened Western rule; suffice to say the Arabs were not of the same mind.
Bell veered between the two a bit, initially siding with Wilson (they were good friends) but then turning her back on him completely and pushing for greater independence for the Arabs when she saw the practical difficulties.
Yet another lobby in India was pushing for Mesopotamia/Iraq to be made into a colony of the Raj that could be settled with Indians from the sub-continent who would then perform essentially the same civilizing function as the British were meant to in other mandates (IIRC German East Africa was also the target of this sort of policy).