So, without the Wahhabism (and somewhat by extension the Saudis), we have a whole lot of butterflies to work with.
1.) There's a possibility that even without the infusion of Wahhabism that the Saudis could have still worked themselves up to be a regional power. While this form of a Saudi polity would be pretty different from our own, it could still come to influence Arabian affairs.
2.) Without the threat of Wahhabism, the Ottomans would be less likely to involve themselves with local Arabian politics. Muhammad Ali, ruler of Egypt, wouldn't campaign in Arabia, which could, in turn, influence his decision to ultimately rise against the Ottomans etc, etc, etc.
3.)Without the Saudis, the interior would remain a tribal patchwork, with the Rashids, Hashemites, Kuwaitis, and the Emirates each claiming their own peripheral segment of the Arabian interior. A clan-oriented conflict would probably default from Rashid-Saudi (as it had been for much of the 18th and 19th centuries) to Rashid-Hashemite. How that shakes out? I couldn't tell you, but without the Saudis, I think the mantle of a mini-Prussia-like unifying force falls to either the Rashids or the Hashemites.
4.) Come post-WW1, the Hashemites would have to deal with fewer threats, potentially ensuring the creation of 3 Hashemite-led states, two of which have large oil reserves and large Shi'a populations, both of which make for trouble in their own special ways...
Those are just a few that come to mind off the top of my head.
