Great Britain got their oil from British Petroleum, which traces its origins back to the Anglo-Persian oil company that was founded before World War I. After WWI and the partition of the Ottoman Empire, BP expanded its operations into Kuwait and Iraq. So Britain did not buy any oil from Saudi Arabia during the WWII era.
Saudi Arabia did not begin to look for oil until after the Standard Oil Company of California (So Cal) began to produce oil on the island of Bahrain, in 1932. Saudi Arabia created a company called Aramco in 1933, to look for oil in Saudi Arabia. So Cal was their first investor and the first oil company to send geologists to look for oil. By 1936, no oil had been discovered yet and Aramco allowed So Cal to sell half of their stake to Texaco. Limited oil production began in 1938, but it was only 1500 barrels of oil a day. Production only increased very slowly during WW II and Saudi Arabia was NOT a major oil producer. In 1948, Exxon was allowed to buy into Aramco, and shortly there after, Mobil was allowed to invest in Aramco too. It wasn't until the early 1950s that the first really large oil field in Saudi Arabia was tapped. The Saudis sold their limited oil production during WWII to Texaco and So Cal, but they were not even in the top ten oil producing nations of the world during the 1940s.
Source: "Aramco and its World : Arabia and the Middle East" edited by Ismail Nawwab, Peter Speers and Paul Hoye