Brannum was born in Sandwich, Illinois in 1910. He attended Maine Township High School in suburban Chicago and went to college at University of Redlands, where he became interested in jazz; after graduation, he played bass in various bands. During World War II, he enlisted in the US Marine Corps and joined a Marine band led by Bob Crosby. After the war, he joined the Four Squires, later moving to Fred Waring and His Pennsylvanians; Waring's group had a regular radio show, where Hugh met Bob Keeshan, an employee at the station who would later hire Brannum for Captain Kangaroo.

Before his time on Captain Kangaroo he hosted a local children's TV series called Uncle Lumpy's Cabin, seen weekday mornings on WABC-TV Ch. 7 in New York City during the 1951 season.

Brannum played a number of characters on Captain Kangaroo from 1955 to 1984 , Mr. Green Jeans, The Professor, Greeno The Clown, The Old Folk Singer, and Mr. Bainter the Painter. His role as Mr. Green Jeans was partly based on stories about a farm kid named "Little Orley" that he told on the radio and on 78-rpm records under the pseudonym "Uncle Lumpy".

Brannum died of cancer in East Stroudsburg, Pennsylvania in 1987.

A long-running but incorrect rumor claims Brannum was the father of musician Frank Zappa, apparently because of a Zappa composition titled "Son of Mr. Green Genes."