Spedding was born at Hamilton, Ontario, in Canada and educated at the University of Michigan and Berkeley where he obtained his PhD in 1929. After working at Cornell from 1935 to 1937, he moved to Iowa State University, where he remained for the rest of his career. He was appointed professor of chemistry in 1941 and director of the Institute of Atomic Research from 1945 to 1968.

In 1942, at the request of Arthur Compton, Spedding and his Iowa colleagues devised new techniques for the purification of the uranium required urgently for the development of the atomic bomb. Their method reduced the price of uranium from $22 per pound to $1 per pound.

After the war Spedding put his new skills to separating the lanthanide elements, an extremely difficult task because of the similarity of their physical and chemical properties. The technique used on uranium, and later successfully applied to the lanthanides, was that of ion-exchange chromatography. A simple example is seen when hard water is allowed to percolate through a column of the mineral zeolite; calcium ions are absorbed by the mineral, which releases its own sodium ions into the water – in effect, an exchange of ions. In the late 1930s more efficient synthetic resins were introduced as ion exchangers.

Spedding passed lanthanide chlorides through an exchange resin that differentially absorbed the compounds present, thus allowing them to be separated. As a result, for the first time, chemists could deal with lanthanoids in substantial quantities.

In 1965 Spedding published, with Adrian Daane, an account of his work in his Chemistry of Rare Earth Elements.