Bawden was born in North Tawton, England, and was educated at Cambridge University, receiving his MA in 1933. From 1936 to 1940 he worked in the virus physiology department at Rothamsted Experimental Station, becoming the head of the plant pathology department in 1940. He was director of the station from 1958 until his death.

In 1937 Bawden discovered that the tobacco mosaic virus (TMV) contains ribonucleic acid, this being the first demonstration that nucleic acids occur in viruses. With Norman Pirie, Bawden isolated TMV in crystalline form and made important contributions to elucidating the structure of viruses and the ways in which they multiply. Bawden's work also helped in revealing the mechanisms of protein formation.