Bullen, the son of Anglo-Irish parents, was born in Auckland, New Zealand; he was educated at the universities of Auckland, Melbourne, and Cambridge, England. He began his career as a teacher in Auckland then lectured in mathematics at Melbourne and in England in Hull. In 1946 he became professor of applied mathematics at the University of Sydney.

Bullen made his chief contributions to science from his mathematical studies of earthquake waves and the ellipticity of the Earth. In 1936 he gave values of the density inside the Earth down to a depth of 3100 miles (5000 km). He also determined values for the pressure, gravitation intensity, compressibility, and rigidity throughout the interior of the Earth as a result of his mathematical studies. From the results on the Earth's density he inferred that the core was solid and he also applied the results to the internal structure of the planets Mars, Venus, and Mercury and to the origin of the Moon.

Bullen conducted some of his early work in collaboration with Harold Jeffreys on earthquake travel times. This resulted in the publication of the Jeffreys–Bullen (JB) tables in 1940.