Born in Newquay in Cornwall, Willcocks began his musical training as a chorister at Westminster Abbey. From 1934 to 1938, he was a music scholar at Clifton College, Bristol, before his appointment as organ scholar at King's College, Cambridge.
With the outbreak of the Second World War, he interrupted his studies in music to serve in the British Army. He won the Military Cross as a temporary captain for his actions on the night of 10/11 July 1944. The battalion suffered over 250 casualties during the night, including the commanding officer and one of the company commanders. This left Willcocks in command of the battalion headquarters. He rallied the men, enabling the battalion to stand firm and reorganise.
Willcocks returned to Cambridge in 1945 to complete his studies, and in 1947 was elected a Fellow of King's College and appointed Conductor of the Cambridge Philharmonic Society. In the same year, he became the organist at Salisbury Cathedral and the conductor of the Salisbury Musical Society. From 1956 to 1974 he was also conductor of the Bradford Festival Choral Society, whilst continuing as guest conductor for their carol concerts into the early 1990s.
From 1957 to 1974 he held the post for which he is probably best known, Director of Music at King's College, Cambridge. He made numerous recordings with the college choir. The choir toured extensively, giving concerts worldwide, as well as garnering further acclaim internationally through television and radio appearances. In 1960, he also became the musical director of the Bach Choir in London.
He held these positions at Cambridge until the 1970s when he accepted the post of Director of the Royal College of Music. In 1971, he was appointed a Commander of the Order of the British Empire (CBE), and was created a Knight Bachelor in 1977. For the 1981 wedding of Prince Charles and Diana Spencer, Willcocks served as director of music and conducted a new piece by William Mathias.
After stepping down from the Royal College, Willcocks resumed conducting and editing scores as his primary activities. On 15 May 2010, a celebration of his contribution to music took place at the Royal Albert Hall in London, where pieces selected by Willcocks were performed by singers who are part of the Really Big Chorus.
His death at home in Cambridge on the morning of 17 September 2015 was announced by King's College later that day.