Bouček was born in the Czech town of Hradec Králové on 8 January 1924, then part of Czechoslovakia. He married Tatiana Rydlova in 1949 and they had one daughter, Jitka. His first works as an entomologist were published while he was at the "Univerzita Karlova" in Prague and he later worked at an agricultural research centre before obtaining his Ph.D. from the Czechoslovak Academy of Sciences, his thesis being a "Revision of Chalcidoidea of Europe". In 1969 he was forced to flee Czechoslovakia as a result of the deterioration of the political situation following the Prague Spring and the subsequent Soviet invasion. He took his family to the United Kingdom where he worked for a short time at the Hope Department at Oxford University before taking up a post at the Commonwealth Institute of Entomology which is located in the Natural History Museum, London where he worked until his retirement in 1989.