Roberts was born in Derby, the son of Edna (Allsop) and John Roberts, an auto mechanic. When he was four, Roberts' family moved to Bath. In Bath, he attended City of Bath Boys' School. As a child he at first wanted to be a detective and then, when given a chemistry set, a Chemist. In 1965 he graduated from the University of Sheffield with a Bachelor of Science degree in Chemistry followed by a PhD in 1969. His thesis involved phytochemical studies of neoflavonoids and isoflavonoids.