Well, on the one hand, the region was already ostensibly British in influence if not in official control, so I'm not sure they would have backed off and let the Italians in after the war. The British are in Aden directly; at the time of the negotiations at Versailles, their "trusted friends" already rule the Hejaz (Hashemites), Nejd (Saud), and Asir (Idrisids). The first wouldn't turn his coat until after the conference made it clear the British were not interested in Arabian independence, and likewise for the timing of the conquests of the second over the first and third. 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. The region was not included in the mandatory territories because it was not of particular interest to the French, and thus not necessary to be partitioned in compromise. Moreover, the Hejaz was incontrovertibly promised to the sharif of Mecca already, where the Levantine coast, Baghdad, and Basra had been left more ambiguous as far as the British and French were concerned. Northern Yemen's position threatened to dominate the key British crown colony of Aden, and moreover, was dominated by the hostile Zaydi; inviting the Italians in would not have been strategically wise, even without foreknowledge of World War 2.
Besides, I'm not sure Italy would have been all that interested. At the peace table in 1919, it cannot be said the Arabian peninsula was of particular interest. In Africa, they were more concerned with British Somalia and Djibouti than Yemen, and were more willing to even make pretensions to bits of Egypt: territorial unity was the watchword for a single grand Italian colony uniting the Horn of Africa. Vague promises had been made and later disregarded about free access to Haifa and Acre in the Levant, but not as much regarding the Arabian shore of the Red Sea. Overall, they were much more concerned with Europe than Africa: Fiume, Trentino and Tyrol, Dalmatia, and the Dodecanese were of crucial interest to them. The tensions resulting from these European demands were quite enough to overthrow the Orlando government on June 19, 1919. It also made the British and French particularly unamenable to just giving the Italians anything. It's worth noting that scant years later, the two big losers in the transition from Sevres to Lausanne are going to be the Greeks and the Italians, and the three big winners the Turks, British, and French; this is not a coincidence. The British and French governments ended up thoroughly sick and tired of the Greeks and Italians over the course of these negotiations, and thus declined to take their considerations into account at the final treaty with Ataturk. They weren't likely to carve out chunks of their own regions of influence, even as indirect as the Arabian peninsula, to give to the Italians.
