Robertson, born the son of a dental surgeon in Cupar, Scotland, was educated at St. Andrews University. After graduating in 1890 he served briefly from 1890 to 1892 in the City Analyst's Office, Glasgow, before entering government service on the staff of the Royal Gunpowder Factory, Waltham Abbey. In 1900 he became chemist in charge, but moved to the Royal Arsenal, Woolwich, in 1907 to serve as superintendent chemist of the research department. In 1921 he became government chemist in charge of the Government Chemical Laboratory in the Strand, London. Robertson remained there until his retirement in 1936 but returned to public service during World War II, which he spent working on explosives at the University of Swansea.

Robertson made a number of advances in the chemistry and technology of explosives. He carried out early work on the decomposition of gun cotton and also improved the process of TNT manufacture. More important was his introduction in 1915 of amatol, a mixture of up to 80% ammonium nitrate to 20% TNT, an explosive more efficient and much cheaper than conventionally produced TNT. It was in fact said of amatol by the director of artillery that it “won the war.”

As a pure chemist Robertson was one of the first to see the value of infrared spectroscopy for determining molecular structure. He consequently used it to explore ammonia and arsine (AsH₃).