Could the House of Sauds take over if the British kept their promise to Kingdom of Hejaz and allowed them to rule over a United Arabia Kingdom like this:
I don’t think that’s a very accurate map. If it is, then no, they can’t as the Hashemites hold too much territory and (presumably) loyalty of tribal forces.
If the map requires editing, it all depends on which way the Brits lean.
The Ikhwan, Saudi tribal allies, lost big when they tried to invade Jordan in the 20s, so I doubt they could conquer even a fraction of that territory. Conquering Yemen, Oman, Lebanon, and Syria, with entirely different local factions and rulers, seems pretty impossible. There's a reason Yemen has remained separate from the rest of the Arabian peninsula since the days of the early Caliphate.