When was the Middle East's vast amount of oil reserves discovered?

Depends on what part of the Middle East. The founder of the Reuters empire had concessions in Persia in 1872 and 1889 to look for oil, but wasn't successful. In the 1890s geologists reported that there might be significant oil in the territory, and at the time the Shahs were leading an luxurious lifestyle that put them into financial debt to Russia. The Iranians sent a guy named Kitabgi to Paris to look for investors in a potential oil concession based on those reports, and found a partner in a British capitalist named William D'Arcy.

D'Arcy gave the Shah 20,000 pounds upfront, 20,000 worth of equities, and 16% of the annual net profits in the concession. By 1904 he had a producing well, but that went dry quickly and in desperation he sought additional capital. The British Government, recognizing the coming shift to liquid fuel in the global naval race, paired D'Arcy with the Burmah Oil Company in 1905 with the promise of lucrative long-term government contracts for fuel. In 1908 the venture struck the first commercially viable well in Persia, and the following year the Anglo-Persian Oil Company went public, with 51% ownership eventually going to the British government in June of 1914.

Meanwhile, similarly promising geological reports were coming out about Mesopotamia, and the capitalists who ignored the chance to get the Persian concession vowed to not be left behind. The Ottomans were cash strapped at the time, and encouraged German financial institutions to compete with Anglo-Iranian for the concession. Soon Royal-Dutch Shell got involved with the Germans through an Armenian oil merchant who owned a chunk of the Turkish National Bank. In 1912 the three formed the Turkish Petroleum Company. Under British pressure, Anglo-Iranian was brought into the mix, and all parties agreed to only operate in the Ottoman empire cooperatively. 11 days after Churchill purchased Anglo-Iranian for the British Government, the Turkish Petroleum Company (of which Anglo-Iranian owned 47.5%), was promised that a formal concession would be granted in Mesopotamia. The problem was that this was the day the Archduke was shot, so the world went to war.

The subject of Ottoman territory was a hot topic among the British and French during the war. The French finally awoke to the need for oil, and established a state oil company that took over the German equity in the Turkish Petroleum Company. This was a concession to them for giving up Mosul in the post-war division of spoils. The next challenge was the legitimacy of the pre-war oil concession, so the British established the state of Iraq and received a new concession in 1925, with drilling beginning in 1927 and oil was struck later that year. At this time American oil companies were brought in for financial stability, and all participants in the newly minted Iraq Petroleum Company signing what has been called the Red Line Agreement, affirming the prewar agreement to only operate cooperatively in the former lands of the Ottoman empire.

For Saudi Arabia, look up a New Zealander named Frank Holmes. After WWI he convinced the Arab shieks along the Arabian side of the Persian Gulf that they might have oil, and got concessions in Bahrain and the neutral lands between Kuwait and Saudi Arabia. The major oil companies all declared that there was no oil in that part of the world, so Holmes couldn't get any financing. It would be Gulf Oil Company that would buy out his concession rights, but, as they had just signed onto the Red Line Agreement through their participation in Iraq, they couldn't operate in Bahrain. The rights were thus sold to Standard Oil of California, while Gulf would pursue oil rights in Kuwait (which was outside the agreement).

Socal hit oil in 1932, while Gulf had to enter into a 50/50 partnership with Anglo-Persian to gain rights in Kuwait by 1933. Meanwhile, Socal was going where absolutely nobody thought there would be any oil, Saudi Arabia, winning a concession in 1933. Both Kuwait and Saudi Arabia would begin producing oil by 1938, and, in the post-war world, they would be developed further.