Siege of Jerusalem (637) - Wikipedia, the free encyclopedia
Interesting link there Penelope.
In answer to the thread title, presuming no other massive US interference in the region (not necessarily realistic) Britain would have not had the issues with Zionist terrorism, so would have been much more successful in bringing the Palestinians to a state of education and competency to run a British style ex-colonial system.
Due to the interruption of WWII it is likely that Britain would, seeing good progress, have accepted the request from the UN to extend its mission to continue to administer the region for some more years. I would think maybe 5, plus 10 more in a supportive role, and providing military security to its borders.
Of course Palestine would develop its own version of parliamentary rule, and have some similarities with India, as well as with Britain. Being small though, it would be able to move faster in education, commerce and development. The gas reserves would probably have been discovered earlier too.
Such a state, on the Med with British supporting security would likely have been commercially and diplomatically influential in the region. The tribal conflicts about it would still fester, but the bloodshed would be more proportionate to how much modern armaments were pushed into the region. As now.
It would likely have a stabilising effect on Lebanon and Jordan.
As to whether we would have had Saddam, the two (three) Iraq wars, the Iran / Iraq wars, well, again that comes down to how much other US / UK / French mischief would be going on in this alternate universe.