During WWI, India paid a subsidy to the Saudis to make sure they would remain neutral. This was needed because the British in Cairo were courting the newly proclaimed King of Hejaz to rebel against the Ottoman empire. After the war, the British installed King Hussein's sons as the monarchs of Jordan and Iran - which was said to have placed a border frontier in between the Saudis and their ancestral lands in Syria.

The money was needed because the Wahhabi Saudis were engaging in a jihad against everyone else, and were thus opposed to the Hashemite king and his sons. But, the money sometimes wasn't enough - the British had to use the Royal Air Force to stop Saudi incursions into Iraq - leading to the Treaty of Mohammera and the Uqair protocols of 1922 which demarcated the borders of Iraq and Nedj (Saudi Arabia), as well as establishing the neutral border zones between Nedj and Kuwait. The message was clear - the British would forcibly defend their mandate territories, and any territory they had treaty obligations with.

Ibn Saud had no choice but to accept British military superiority - however, he always distrusted them, which led to the later warm relations with the United States despite Britain's perceived authority in the area. For their part, the British allowed the Saudis to do what they wanted when it didn't interfere with treaty obligations. King Hussein failed to sign a formal agreement between Hejaz and Britain, much to the relief of the British when he later proclaimed himself Caliph in 1925 and invited the Saudi invasion. For their part, the Saudis had stopped receiving their wartime subsidy, so thus had no economic incentive to keep their jihad under wraps. They naturally expanded into whatever space wasn't under British protection.

The only true sticking point was in Jordan, as nobody had established the boundary between Hejaz and the new mandate territory. The British wanted to include the port of Aqaba into the mandate due to it's strategic significance, but Hejaz effectively controlled it due to their performance in the war. When Ibn Saud forced Hussein to flee from Mecca, it was to Aqaba that Hussein fled - funneling recruits and materiel from his son Abdullah to his other son, Ali, who had taken over leadership after Hussein's abdication. Saud was going to take Aqaba and the nearby railroad town of Mann, except the British finally stepped in - forcing Hussein to go to Cyprus and warning Ibn Saud that they considered both cities to be within Jordan. The border would be finalized satisfactorily later by the successors of the two combatant leaders.

Well up until ww1 what we know know as Saudi Arabia was part of the ottoman empire. Also most of the territories surrounding it were British mandates post ww1 and weren't formally part of the empire and only under British control for a few years.