Dudley-Smith was born on 26 December 1926 in Manchester, England, to Phyllis and Arthur Dudley-Smith. His father was a schoolteacher. He was educated at Tonbridge School before studying maths and then theology at Pembroke College, Cambridge. After graduating in 1947, he began his ordination training at Ridley Hall, Cambridge. He was ordained deacon in 1950 and priest in 1951 by Christopher Chavasse, the Bishop of Rochester.

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