David was born in the Wandsworth area of London to Ian Frederick Lettsom Elliott and his wife, Florence Elizabeth Vere Fison. He had an older sister and a twin brother who died shortly after their birth. His father was the representative of US Steel in Europe who, in the years immediately before World War II, had participated in the management of the International Steel Cartel and, later, was a representative of the United States on the international control authority for the German Ruhr.

David’s mother died on 1 March 1927. She was a sister of James Frederick Lorimer Fison, a graduate of Christ Church who died of wounds in 1917.

David was educated at Rugby where he was a Scholar and Matriculated in 1937.

He was a good cricketer and played for Suffolk in the 1938 season. With his sister, Diana, he spent much time at Stutton Hall, Suffolk, the home of their Fison grandparents.

He joined the 1st Battalion Welsh Guards at Midsomer Norton on 5 October 1941 from a training battalion.

He was killed in the battle for Monte Cassino on 11 February 1944.

He is buried in the Cassino War Cemetery Plot XIX. H. 11.

His sister, Diana Clavering Elliott was born at Stutton Hall on 13 August 1917. Her father was serving with the Suffolk Regiment in France. After his demobilisation, the family settled in Putney where the twins were born in 1919. She gave up reading English at Lady Margaret Hall, Oxford, to marry in October 1939 the Rev John Collins, then Dean of Oriel and later Chairman of the Campaign for Nuclear Disarmament. She famously remarked, "I would rather be married to the Chairman of CND than to the Archbishop of Canterbury."

She was appointed DBE in 1999 and died in May 2003.