Given that more than a few of the Middle-East's problems (in particular with regards to the Israel-Palestine conflict) can be traced back to decisions made by the British and French in the aftermath of WW1. Leaving aside the how for the moment, what would have happened if the British were put in charge of administering what is now Syria and Lebanon, whilst France was put in charge of Iraq, Palestine and Jordan?
Would the British have split Lebanon from Syria? Would the French have given Transjordan to the Hashemites? Might the British have installed the Hashemites in Syria?
That's strange, I was thinking of posting this scenario myself, although I was going to keep Iraq British, and only switch Syria+Lebanon for Palestine+Transjordan. The British might have an easier time than OTL between the wars, as I think they have the capability to deal with the Druze uprisings for example, better than the French did. It was the Jewish vs Arab conflict which the British struggled with most, and I don't think France's colonial attitude would have won it favours with either side. Britain's more liberal view on self-government could have led to a reasonably stable independent Syria, while I imagine Lebanon might have been carved off anyway.
In France's new territory, if they treat it as a colony with only limited local power, then they will struggle to contain uprisings in Iraq, far from any secure French territory. Britain at least had Kuwait, but France will have to improve communications between Iraq and the Med, or face losing effective control over northern Iraq. Kurdish independence looks more likely, especially with British support. When (or if) WW2 comes, and I think it will, if there is still an equivalent to Vichy France, the Middle Eastern situation will be very interesting whatever state the region is in, Anglo-French rule or not. This could make a good TL.