Milligan was born, the son of an Irish-born father, Captain Leo Alphonso Milligan, MSM, RA, who was serving in the British Indian Army. His mother, Florence Mary Winifred Kettleband, was born in England. He spent most of his childhood in Rangoon (Yangon), capital of Burma (Myanmar). He was educated at the Convent of Jesus and Mary, Poona, and St Paul's Christian Brothers, de la Salle, Rangoon.

He lived most of his life in England and served in the British Army, in the Royal Artillery in World War II.

During most of the late 1930s and early 1940s Milligan performed as an amateur jazz vocalist and trumpeter before, during and after being called up for military service in the fight against Nazi Germany, but even then he wrote and performed comedy sketches as part of concerts to entertain troops.

After his call-up, but before being sent abroad, he and fellow musician Harry Edgington (nicknamed Edge-ying-Tong which gave birth to one of Milligan's most memorable musical creations, the Ying Tong Song) would compose surreal stories, filled with puns and skewed logic, as a way of staving off the boredom of life in barracks.

During World War II he served as a signaller in the 56th Heavy Regiment Royal Artillery, D Battery, as Gunner Milligan, 954024 with the First Army in the North African campaign and then in the succeeding Italian campaign. He rose to the rank of Lance Bombardier and was about to be promoted to Bombardier when he was wounded in action in Italy.

Subsequently hospitalised for a mortar wound to the right leg and shell shock , he was demoted by an unsympathetic commanding officer (identified in his war diaries as Major Evan 'Jumbo' Jenkins) back to Gunner. It was Milligan's opinion that Major Jenkins did not like him due to the fact that Milligan constantly kept the morale of his fellow soldiers up, whereas Major Jenkins' approach was to take an attitude towards the troops similar to that of Lord Kitchener.

An incident also mentioned was when Major Jenkins had invited Gunners Milligan and Edgington to his bivouac to play some jazz with him, only to discover that the musicianship of the aforementioned gunners was far superior to his own ability to play the military tune 'Whistling Rufus' (albeit badly).

After his hospitalisation, Milligan drifted through a number of rear-echelon military jobs in Italy, eventually becoming a full-time entertainer. He played the guitar with a jazz and comedy group called The Bill Hall Trio in concert parties for the troops.

After being demobilised, Milligan remained in Italy playing with the Trio but returned to England soon after. While he was with the Central Pool of Artists (a group he described as composed "of bomb-happy squaddies") he began to write parodies of their mainstream plays, that displayed many of the key elements of what would later become The Goon Show with Peter Sellers, Harry Secombe and Michael Bentine.

Milligan returned to jazz in the late 1940s and made a precarious living with the Hall trio and other musical comedy acts. He was also trying to break into the world of radio, as either a performer or as a script writer. His first success in radio was as writer for comedian Derek Roy's show.

Milligan soon became involved with a relatively radical comedy project, The Goon Show. Known during its first season as Crazy People, or in full, "The Junior Crazy Gang featuring those Crazy People, the Goons!", the name was an attempt to make the programme palatable to BBC officials by connecting it with the popular group of comedians known as The Crazy Gang.

Milligan was the primary author of The Goon Show scripts (though many were written jointly with Larry Stephens, Eric Sykes and others) as well as a star performer.

Milligan had a number of acting parts in theatre, film and television series; one of his last screen appearances was in the BBC dramatisation of Mervyn Peake's Gormenghast, and he was (almost inevitably) noted as an ad-libber.

One of Milligan's most famous ad-lib incidents occurred during a visit to Australia in the late 1960s. He was interviewed live on air and remained in the studio for the news broadcast that followed (read by Rod McNeil), during which Milligan constantly interjected, adding his own name to news items. As a result, he was banned from making any further live appearances on the ABC.

The ABC also changed its national policy so that talent had to leave the studio after interviews were complete. A tape of the bulletin survives and has been included in an ABC Radio audio compilation, also on the BBC tribute CD, Vivat Milligna [sic].