In 1886 (circa) Henry Priestley migrated to Sydney, Australia with his parents. After completing his undergraduate education Priestley enrolled at the University of Sydney. First he studied arts (1902), then engineering (1903), science (1904) and finally medicine (1906). He graduated with first class honours in medicine and won the University medal. Priestley was awarded the Beit memorial Fellowship in 1910 and chose to study nutrition at London's Lister Institute of Preventive Medicine. He returned to Australia in 1913 to work as a bacteriologist at the Australian Institute of Tropical Medicine, Queensland. His research covered haematological aspects of hookworm infestation, and studies on many disease including trachoma and typhoid fever.

In 1918 Henry Priestley returned to New South Wales to work at the University of Sydney. His tenures included lecturer and chief physiology demonstrator, associate professor of biochemistry (1921), acting professor of physiology (1928), Head of the newly formed Department of Biochemistry (1939), Dean of the Faculty of Science (1946) and Fellow of the University Senate (1948). His work at the University saw the establishment of a BSc degree in Medicine, and a research training program for medical students. Other outside appointments held by Priestley include foundation fellow of the Royal Australasian College of Physicians, member of the Linnean Society, and advisor to many committees of the National Health and Medical Research Council (NHMRC).