Radford was born in Manchester in 1793. He was apprenticed to his uncle, William Wood, a surgeon at the Manchester and Salford Lying-in Hospital in 1810. Later Radford became partner to Wood and eventually took over his practice. Radford completed his medical education in London, at St Thomas's and Guy's hospital. He received his medical qualifications in 1817. In 1818 he became man-midwife for the Lying-in hospital, and was promoted to surgeon-in-ordinary in 1825 surgeon extraordinary in 1834, and consulting physician in 1841, a position he held until his death.
Radford soon built up a considerable reputation as an obstetrician. In 1839 he took a MD from the University of Heidelberg. Radford was a lecturer at Mount St School, and after this closed, at Manchester Royal School of Medicine. He published a number of works on obstetrics including Observations on the Caesarean section a, craniotomy and other obstetric operations (1865). Radford did much to improve the status of the Lying-hospital, raising money for a new hospital in Quay St which opened in 1856. In 1853 he also presented his remarkable collection of books on obstetrics and gynaecology to the hospital (in 1927 this passed into the custody of the University of Manchester library).