A History of the Modern middle East by William L. Cleveland is an incredibly good read about what followed when the ottoman empire collapsed. Anyway on to your question, during world war 1, the region that would become Saudia was mainly Nejd, Haasa, shammaar and Hedjaz. The regions that were not Hedjaz were split between various families. The house of Saud and the house of Rashid being two very notable ones. Hedjaz on the other hand was part of the ottoman sultanate, during world war one the Sharif Hussein bin Ali of Hedjaz talked to the British Commissioner McMahon in Egypt about revolting against the ottomans, in exchange for support in the formation of an arabian kingdom being formed in Syria, Iraq, Jordan, and Arabia. Although agreed upon the McMahon correspondence was incredibly vague and in the end, after a short lived Kingdom of Damascus was crushed, The Sharifs sons were placed in rule in Iraq and Jordan (where the dynasty still rules in Jordan.)

While this was happening, Ibn Saud was unifying the tribes, as well as using Wahabi militia named the Ikhwan, to consolidate power in Saudia, as well as marrying into all other tribes. He was able to quite easily overwhelm and conquer Hedjaz.

These Ikhwan was a huge part of how he consolidated and stabilized himself, and he would later turn his back on them.

The money from oil did not really play a part, and actually they Saudies believed for a long time that they would not find oil in Arabia, it was only by accident, when searching for water did they find oil. Which did help in getting themselves into the American sphere of influence protecting themselves from any British intervention that we see in Iran.

The emir states in the gulf were largely protected by Britain, but i'm not quite familiar on that topic.