Nagalingam was born on 25 October 1893. He was the son of Chellappah and Meenachchi from Urumpirai in Northern Province of Ceylon. He hailed from a distinguished family and had four eminent brothers: C. Suntharalingam, a member of parliament and government minister; C. Panchalingam, a medical doctor; C. Amirthalingam, Director of Fisheries; and C. Thiagalingam, a leading lawyer. Nagalingam was educated at St. John's College, Jaffna and Royal College, Colombo where he excelled in studies and sports and won the De Soysa Science Prize. Thereafter he entered Ceylon Law College, qualifying as an advocate in 1917.
After Nagalingam was called to the bar, he practised law in Colombo until 1937. In 1938 he was appointed as an Additional District Judge of Colombo. He was appointed District Judge of Kandy in 1941 and acting Attorney General in 1946. He was also appointed King's Counsel in 1946. In 1947 he became acting Legal Secretary, one of the three official members of the State Council, heading committee of justice with ministerial rank in the Board of Minister. He was also called to the bench as an acting Puisne Judge and was confirmed judge of the Supreme Court in 1947, the first Tamil to hold that post. He was later appointed Senior Supreme Court Judge and acted as Chief Justice on a number of occasions. In this capacity he was called upon to briefly serve as acting Governor-General in 1954 whilst Lord Soulbury was out of the country.
Nagalingam was one of the founders of the Hindu Educational Society which founded the Colombo Hindu College in 1951.