  We study the Anomaly flow on $2$-step nilmanifolds with respect to any
Hermitian connection in the Gauduchon line. In the case of flat holomorphic
bundle, the general solution to the Anomaly flow is given for any initial
invariant Hermitian metric. The solutions depend on two constants $K_1$ and
$K_2$, and we study the qualitative behaviour of the Anomaly flow in terms of
their signs, as well as the convergence in Gromov-Hausdorff topology. The sign
of $K_1$ is related to the conformal invariant introduced by Fu, Wang and Wu.
In the non-flat case, we find the general evolution equations of the Anomaly
flow under certain initial assumptions. This allows us to detect non-flat
solutions to the Hull-Strominger-Ivanov system on a concrete nilmanifold, which
appear as stationary points of the Anomaly flow with respect to the Bismut
connection.
