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Davenport was born and educated in Accrington, where his gifts for mathematics and chemistry were early recognised. At the age of sixteen he was awarded scholarships to attend Manchester University (1924-1927), where he won first-class honours and the respect of such tutors as L. J. Mordell and E. A. Milne. The latter encouraged him to enter for a scholarship at Trinity College Cambridge, where he read for the Tripos (1927-1929), began research with J. E. Littlewood, won the Rayleigh Prize (1931) and a Trinity Fellowship (1932-1937). Another important event of this period was Davenport's acceptance of an invitation by H. Hasse to stay with him in Marburg, a visit which enabled Davenport to meet several distinguished German mathematicians and to learn the language thoroughly.

Davenport's first university post was a return to Manchester as Assistant Lecturer (1937-1941) and in 1941 he was appointed to his first Chair at University College of North Wales, Bangor. In 1945 Davenport, who had married Anne Lofthouse in 1944, moved to University College London as Astor Professor. His first American visit, to Stanford in 1947-1948, was followed by many later transatlantic journeys of varying duration. While at the Mathematics Department of University College Davenport launched the new journal Mathematika in 1953. In 1958 he was elected Rouse Ball Professor of Mathematics at the University of Cambridge and Fellow of Trinity College, Cambridge, and held those positions until his death in 1969.