Final answer:
Winston Churchill was the British Prime Minister at the time of Gandhi's 'Quit India' speech in 1942, during World War II.
Step-by-step explanation:
The British Prime Minister at the time of the "Quit India" speech in August 1942 was Winston Churchill. This was a period during which World War II was ongoing, and Churchill had been in power since 1940 after the resignation of Neville Chamberlain. The Quit India speech, led by Mahatma Gandhi, was a significant moment in the Indian independence movement, calling for the end of British rule in India. Churchill, a lifelong British politician, who had been warning of the dangers posed by Adolf Hitler, leader during a time of war and tumultuous global relations.