Bancroft was born in Saskatoon, Saskatchewan, the son of an accountant, but grew up in Winnipeg, Manitoba where he attended Kelvin High School.   
He graduated from the University of Manitoba in 1963, subsequently earning an MSc in chemistry (1964) from the same institution. Later that year he went to the University of Cambridge, England to study for a PhD at Corpus Christi College. He worked under the supervision of A.G. Maddock on the development of Mössbauer spectroscopy, obtaining his PhD in 1967. Over a 20-year period Bancroft would become one of the world's leading experts in Mössbauer spectroscopy, publishing more than 80 papers, a major review and an authoritative textbook in the field.