  This is the second of two papers introducing and investigating two bivariate
zeta functions associated to unipotent group schemes over rings of integers of
number fields. In the first part, we proved some of their properties such as
rationality and functional equations. Here, we calculate such bivariate zeta
functions of three infinite families of nilpotent groups of class 2
generalising the Heisenberg group of three by three unitriangular matrices over
rings of integers of number fields. The local factors of these zeta functions
are also expressed in terms of sums over finite hyperoctahedral groups, which
provides formulae for joint distributions of three statistics on such groups.
