His doctoral students included Gilbert Baumslag, László Kovács, Michael Newman, and James Wiegold.

In 1962 he migrated to Australia to take up the Foundation Chair of the Department of Mathematics within the Institute of Advanced Studies of the Australian National University (ANU), where he served as head of the department until retiring in 1975.

He was an invited speaker of the International Congress of Mathematicians in 1936 at Oslo[5] and in 1970 at Nice.

[6] The Australian Mathematical Society awards a student prize named in his honour.

[7] The group-theoretic notion of HNN (Higman-Neumann-Neumann) extension bears the names of Bernard and his wife Hanna, from their joint paper with Graham Higman (who later supervised the PhD of their son Peter).