Trans-Jordan was given to Abdullah as a bit of a reward for not getting involved in the fighting in Syria, part of the general awarding of booby prizes to the Arabs which had been promised so much during the war. If there is no "Jewish homeland" to worry about, would the British give him the entire area, including the more populous - and very religiously significant to _Christians_ west-of-the-Jordan bits? I'd think they'd rather spin it off as a separate colony, which probably post world war II gains it's independent in fairly short order during the general 1947-1960-something decolonization scramble.
Might even get some sort of initially more-or-less democratic government, but I'm doubtful it stays so for long. If Nasser isn't butterflied away, or if we get a similar strong unify-the-Arabs boss man in Egypt, geographically contigous and non-Monarchic Palestine will be first on the unionist agenda, which might even have popular support in Palestine. (What did pre-war Palestinians think of Egyptians, BTW? Beating Heart Of The Arab World or a bunch of disorderly fellahim?) If it happens, the aforementioned geographic continuity means the union might stick - which may be settings things up for a _big_ war later on...
(I think in the absence of Israel, the conflict between traditional monarchies and modernizing, leftist or *Baathist "republics" will be, if anything, sharper: no israel as a common enemy - although, I will admit, that never seemed to restrain people much - no diversion of energies into wars vs Israel, and Egypt is no longer geographically cut off from the Arabian peninsula. )
No Israel means no lost wars with Israel, which means less humiliation for the Arabs, which may mean positive things for the development of middle eastern democracy over the long run, if a series of nasty wars in failed attempts to "unify the Arabs" don't mess things up. (And people will have less of a perception of Arabs being unable to fight their way out of a paper bag even if given an exacto-knife. Or am I assuming too much here? There seemed to be a willingness to be impressed by the prowess of Iraq's armies in 1990, unless that was all last-minute propagandizing).
Without the Palestinian conflict spilling over into Lebanon, will that country's fragile balance be preserved, or is it still doomed to civil war as the Christian % of the population decreases and the Shiites push for a bigger slice of the pie?
Our foisting the Shah on the Iranians probably is independent of the existence or otherwise of Israel, so he probably still gets the nod from the US and gets to offend a great many Shi'a clerics. Butteflies are big enough that it's hard to say whether we get an iranian revolution like ours, and when it might happen.
Without the wars of OTL, we probably don't get the oil embargo, but the big oil producers getting together to flex their muscles is probably inevitable: oil prices probably rise substantially over the 70's and perhaps 80's, but it might be gradual enough that the panicky attitude of OTL is not duplicated. (Also, the USSR's oil sales will not follow the "boom-and-bust" cycle of OTL, with possible effects on the empire's stability).
No Israel, the US lacks a stable ally in the area in the later parts of the century. (I don't think the US really considered Israel an ally for the first couple of decades, at least). Does the US try harder to find and build up friends in the area? ("Yes, Saddam is a S.O.B. But he's _S.O.B.")
Bruce