  By adapting a technique of Molchanov, we obtain the heat kernel asymptotics
at the sub-Riemannian cut locus, when the cut points are reached by an
$r$-dimensional parametric family of optimal geodesics. We apply these results
to the bi-Heisenberg group, that is, a nilpotent left-invariant
sub-Rieman\-nian structure on $\mathbb{R}^{5}$ depending on two real parameters
$\alpha_{1}$ and $\alpha_{2}$. We develop some results about its geodesics and
heat kernel associated to its sub-Laplacian and we illuminate some interesting
geometric and analytic features appearing when one compares the isotropic
($\alpha_{1}=\alpha_{2}$) and the non-isotropic cases ($\alpha_{1}\neq
\alpha_{2}$). In particular, we give the exact structure of the cut locus, and
we get the complete small-time asymptotics for its heat kernel.
