He was born in Vancouver, British Columbia, Canada, to Scottish parents, one of six children of John H. and Mary Littleson McKellar. He studied mathematics and physics at the University of British Columbia, graduating in 1930. He began graduate studies at the University of California, being awarded his M.S. in 1932 and a Ph.D. the following year. Applying to the United States National Research Council, he was awarded a post-doctoral study program for two years at MIT.
In 1935 he joined the Dominion Astrophysical Observatory, where he performed research into astrophysics. He was married to Mary Crouch (b. June 3, 1911, d. Nov. 30, 2000) in 1938, and the couple bore two children, Andrew Robert William (Bob) (b. March 28, 1945), and Mary Barbara (b. Nov. 1, 1946) (McKellar) Bulman-Fleming. During World War II he served with the Royal Canadian Navy, in the Directorate of Operational Research.
Following the war, from 1952 until 1953, he was visiting professor at the University of Toronto department of physics. Between 1956 and 1958 he served as president of the Astronomical Society of the Pacific, then in 1959 he became president of the Royal Astronomical Society of Canada for a year. He continued working at the DAO up until four days before he died in Victoria, British Columbia from complications due to lymphoma contracted during his service in the Navy during the War.
He was noted for his work in molecular spectroscopy. Among his contributions was the first estimation of the temperature of interstellar gas (and therefore deep space) as 2.4° K based on the excitation of CN doublet lines, and finding evidence for the carbon-nitrogen nuclear cycle as the energy source for carbon stars. (The temperature estimate was subsequently confirmed with the discovery of the cosmic microwave background radiation, which has a measured temperature of 2.725 °K.) During his career he was the author (or co-author) of 73 scientific publications.