Bradley was educated at the University of Cambridge where he earned Bachelor of Arts, Master of Arts and PhD^[15] degrees in genetics from Trinity College, Cambridge gained while working in the laboratory of Martin Evans.^[9][7]

Following his PhD, Bradley was appointed assistant professor at Baylor College of Medicine, beginning in 1987 where he was also a Searle Scholar in 1988.^[16] Bradley was appointed a Howard Hughes Medical Institute Investigator in 1993 and director of the Wellcome Trust Sanger Institute, from October 2000 (preceded by John Sulston) to April 2010, succeeded by Michael Stratton.

Bradley won a 1994 DeBakey Award^[17] and was elected a Fellow of the Royal Society (FRS) in 2002. His certificate of election reads:

"Allan Bradley has made important contributions to the technique of mutation of endogenous genes in mice, an approach that has opened up a new era of research in biology. It is impossible to open an issue of a major journal nowadays without coming across an article that describes the consequences of mutating an endogenous gene in mice. The generation of these mice is based on concepts and techniques that can be traced back to experiments performed and published by Bradley fifteen years ago. In the years since, he has not only used ES cell technology to provide key information on the functions of many genes including several important tumour suppressor genes, but has also continued to improve and develop the techniques, technology, and tools for genetic manipulation in the mouse. Today, mice can be generated with changes as subtle as an alteration in a single nucleotide or as extensive as the deletion of millions of base pairs. These alterations will gain increasing importance in genetic experiments aimed at understanding the function of genes in the mammalian genome in the post genome era."^[10]