Ransome was a member of a leading Manchester medical family, being the grandson of John Atkinson Ransome and son of Joseph Ransome. He was educated at Trinity College Dublin and at Cambridge, where he took a first in natural sciences. He studied medicine at St George's Hospital, London, and received the MB in 1858. He was awarded the MD in 1869. Ransome originally worked in general practice in Bowdon, near Altrincham, Cheshire, but he is better known as one of the leading public health experts in Victorian Manchester. In 1874 he was appointed lecturer in hygiene at Owens College, and was later promoted to a chair. He was also a consulting physician at the Manchester Hospital for Consumption. He was an active member of the Manchester and Salford Sanitary Association.
Ransome's special interest was diseases of the lungs especially TB. Ransome laid particular emphasis on environmental factors in promoting TB, particularly poor housing. He published widely on this subject, including a study of mechanism of respiration, Stethometry 1876) and his invention of the stethometer won him a fellowship of the Royal Society. He also published State medicine in England (1877), Prognosis of lung disease (1882), and Treatments of Phthisis (1896). His collected papers were published in 1915.