Born in Birkenhead, Thelwell spent World War II in the East Yorkshire Regiment, having signed up at the age of 18 in 1941, and was art editor of an army magazine in New Delhi, India.
His first published cartoon, in the London Opinion, was an Indian subject.
In 1944, he took evening classes in art at Nottingham Art School. A fellow art student, Rhona, became his wife in 1949. They had one son and one daughter.
After Nottingham, he took a degree at Liverpool College of Art, then in 1950, he took up a post teaching design and illustration at Wolverhampton College of Art, but gave this up to work freelance in 1956.
He became a contributor to the satirical magazine Punch, who first published his work in 1952, beginning a 25-year relationship that resulted in more than 1,500 cartoons, of which 60 were used as front covers. He also worked as political cartoonist for the News Chronicle from 1956 until the paper closed in 1960.