While Wahhabism may simply be a local group of troublemakers with some influence in Rashid Arabia, I don’t this will stop modern Islamic terrorism from developing, Arab nationalism will still fail, militant Islam will still be a useful tool against USSR, the Rashid will make the same alliance as the Saudi did with USA. The Muslim Brotherhood develop fully independent from Wahhabism and Hamas are part of it, the Syrian Hama Uprising also had them as the main actor.
But we wouldn’t see the Rashid as active founders of reactionary Islamism as the Saudis, while rich they would likely lack some of the Saudi money, and without Hejaz they won’t have as great need to prove their piousness, so instead we will likely see a more ad hoc support of Islamism from the oil rich Arabic monarchies (which may include Iraq), which will likely also lead to political Islam being even more diverse than in OTL.
So we will likely see Islamic terror as pretty local things connected to distinct countries and their diasporas, like what we saw with Algeria and France in the 90ties.
The thing is that the Saudis used their oil money to fund Wahhabist schools of thought around the Muslim world for decades. They were already kinda nutters and I doubt the Rashidi or their Ottoman allies would let them be around given their historical ties to the House of Saud.
Meanwhile, it'd all depend on what the Hashemites would do. Without the House of Saud, the British would have to remain with them and the Hashemites was wanting to unite all of Arabia under them. I don't know the historical relations between them and the Hashemited, but given how the Rashidi were historical allies of the Ottomans, that would lead to trouble. Granted, a stronger Rashidi would mean either the Hashemites would want to avoid trouble with them or force the British to provide more assistance.
It could lean to a unification war for Arabia between the Hashemites and Rashid. The Al-Khurma dispute did not happen. Who would win would depend on who Europe would be more keen to back up. The British fought against the Rashid and thus they would not want them to win though Turkey is the wild card here. If Turkey maintains ties with Jamal Shammar, then they'd easily crush Hejaz and the holy land would be tied under the House of Rashid. They would likely use the oil money to either work on their own things or promote more traditional and orthodox Islam. Alternately, it could be a more progressive one of the Rashidi heirs studied in Turkey and were shaped more by their ideals.
However, if Turkey neglects the House of Rashidi and the latter stills goes to conflict, this would likely spur a conflict where the British and possibly French would have to commit more forces to help the Hashemites if at all possible.
Alternately, the House of Rashidi does not really care much on nationalism besides defending itself and just works as a neutral state.