Ransome was born in Norwich, and apprenticed to a surgeon in King's Lynn. He completed his medical education in London. After unsuccessful tempts to set up a practice in Suffolk, he moved to Manchester and was appointed honorary surgeon at the Infirmary in 1806. Ransome lectured in anatomy at the Lit & Phil. and was its secretary from 1810-1820. He later played a leading role with Thomas Turner in the establishment of the Manchester Medical School, and he lectured in surgery there. Ransome had a large and successful private practice. Ransome was one of the surgeons who treated William Huskisson following his accident at the opening of the Manchester and Liverpool Railway in 1830, and wrote an account of the injury in the North of England Medical and Surgical Journal 1830.