Mellish was born in Rondebosch, Cape Town where he attended Wynberg Boys' High School, Rondebosch Boys' High School, and SACS. During World War I, he served in South-West Africa as a gunner with the Cape Town Highlanders in the South African Heavy Artillery from 1914 to 1915. On 15 May 1915 he sailed on the "SS City of Athens" to Europe where he received the Military Cross for action at Ypres in 1916. He undertook officer training in England, followed by a stint at the Royal Artillery Cadet School before his promotion to 2nd Lieutenant (and later acting Lieutenant). He then served on the Western Front with the South African Heavy Artillery where he was wounded in the left leg in April 1918. He relinquished his commission on 1 June 1919.