The anecdote comes from Sir Sherard Cowper-Coles, the former British ambassador to Saudi Arabia, who was with the Queen during a visit from then-prince Abdullah.
… in September 1998, Abdullah had been invited up to Balmoral, for lunch with the Queen. Following his brother King Fahd's stroke in 1995, Abdullah was already the de facto ruler of Saudi Arabia. After lunch, the Queen had asked her royal guest whether he would like a tour of the estate. Prompted by his Foreign Minister, the urbane Prince Saud, an initially hesitant Abdullah had agreed. The royal Land Rovers were drawn up in front of the castle. As instructed, the Crown Prince climbed into the front seat of the Land Rover, with his interpreter in the seat behind. To his surprise, the Queen climbed into the driving seat, turned the ignition and drove off. Women are not – yet – allowed to drive in Saudi Arabia, and Abdullah was not used to being driven by a woman, let alone a queen. His nervousness only increased as the queen, an Army driver in wartime, accelerated the Land Rover along the narrow Scottish estate roads, talking all the time. Through his interpreter, the Crown Prince implored the Queen to slow down and concentrate on the road ahead.
Cowper-Coles says that while it's bad form to repeat what the Queen says in private to her trusted subjects, this was "too funny not to repeat." The story's supposedly been confirmed by the late King, but even the Queen's skills weren't enough to get him to overturn his county's ban on women driving.