Well, with the obvious disclaimer that it is really impossible to say…
Iraq - No change. Attempts to maintain the Hashemite dynasty would likely fail in this alternate world, just as they did in the real one. Arabia is separated from the major cities of Iraq by considerable expanses of desert and would have had neither the time nor the military force to re-assert Hashemite rule after a coup.
Jordan - Minor, but probably insubstantial change. Jordan would likely feel more secure, but they already had their own dynastic powerbase and, again, there is some distance between major centers in sparsely populated Arabia and Jordan.
Israel - No substantive change. The Hashemites in Arabia would be as conservative, probably more so, than the Jordanians ( who fought in '48 and '67 and would have fought in '73 if they had had more modern air defenses installed in time ). Though doubtless more moderate than the Saudis, it would only be a matter of degree and really the Saudis have always been peripheral at best in this conflict. Their major contribution, funding, would have likely still have been available for the PLO, but perhaps not for some of the most extremist factions.
OPEC - No change. Hashemite Arabia would have just as much motive to help organize and take a leadership role in such an organization as the Saudis did.
Relations with the West - Minor change. Saudi Arabia has been staunchly pro-Western. Hashemite Arabia would almost certainly ( barring unforseen but quite possible internal reaction ) have remained so as well. The only substantive difference might have been that a HA would likely have been more British and less U.S.-oriented than SA.
Role as regional power-broker through economic influence - No change.
Role in spreading and promoting religious fundamentalism - Ah, here we would have change. As guardians of Mecca and Medina it is a foregone conclusion that the Hashemites would have been somewhat conservative. But they weren’t anywhere near as strict as the Wahhabis. However…The Hashemites did not enjoy the tribal loyalty in the Nejd that they did in Hijaz. Wahhabism would have either gone underground or simmered uneasily under Hashemite rule. Tribal rebellions in the east would have been likely ( successful? impossible to say ). So I imagine ObL’s would have still been possible. Still, funding of religious conservative movements would have been far less thorough, if they even occurred at all. Two likely scenarios:
A) Turn towards fundamentalism anyway as a counter to anti-royalist leftism.
B) Internal war with religious fundamentalists
Would this mean fundamentalism would be less of a problem today? Quite possibly, but only by a matter of degree. Remember that Saudi Arabia is not the primary ideological font of modern fundamentalism, despite the erroneous labeling of fundamentalists generally as “wahhabis” in some areas. Far more important worldwide has been the influence of Indian, Iranian, and Egyptian theologians and from a practical point of view the events in Pakistan, Afghanistan, Algeria, and Egypt. Radical jihadists-slafists would still be an issue. How much of one in comparison is pretty tough to say.
MHO.