Neilson was the son of Louis Neilson and Anne Perry Rodgers. His frandfather, Admiral Christopher Raymond Perry Rodgers, was president of the Naval War College and was also superintendent of the Naval Academy at Annapolis. Following in the family tradition he attended the Naval Academy, graduating in 1905. While in school he was a champion fencer and boxer. He then entered the Navy but left in 1908 because of ill health and to study painting.
Neilson studied at the Boston Museum of Fine Arts School and in New York at the Art Students League under J. B. Bridgeman and W. M. Chase. In Paris he studied at the Academie Julien under J.P. Laurens, and at the Ecole des Beaux Arts, the Grand Chaumiere and Academie Colorrossi under Simon, Caro Del Vaile and Richard Miller. He won silver medals at the Paris Salon in 1914 and at the Panama Pacific Exposition in 1915. THe French government purchased his work "Le Chapeau Noir" for the Luxemburg Collection. In 1907 he married Mary S. Park.
Neilson was an instructor in the life class at the National Academy (1927-34), served on the school committee (1940-41) and as recording secretary (1937-46). He also taught at the Art Students League (1926-27) and served on the council of Allied Artists of America.
From the Connecticut Academy of Fine Arts he won the Howard Penrose Prize in 1944 and in 1949 the Alice Collins Dunham Prize for "Holiday Portrait". In 1941 he exhibited society portraits, with many of the figures in sporting attire, at the Newhouse Galleries.
After World War II he was commissioned by William D. Pawley, the organizer of the Flying Tigers, to paint 27 portraits of men who had died in combat flying missions for the Flying Tigers. These paintings were exhibited in 1945 at the Natural History Building, U.S. National Musuem and the exhibition was called "Americans Valiant and Glorious".
