Primrose was born in Glasgow and studied violin there and, later, at the Guildhall School of Music in London. From there he moved to Belgium to study under Eugene Ysaye who encouraged him to take up the viola instead. He found success on the instrument, playing in the NBC Symphony Orchestra under Arturo Toscanini and touring Europe and America as a soloist. In 1944 he commissioned a viola concerto from Bela Bartok.
Later in his life, Primrose became a noted teacher, writing several books on viola playing and teaching widely in the USA and Japan. He died from cancer in Provo, Utah.