  In Part I of this series, we introduced the Spherical Collapse (SC)
approximation in Lagrangian space as a way of estimating the cumulants $\xi_J$
of density fluctuations in cosmological Perturbation Theory (PT). Within this
approximation, the dynamics is decoupled from the statistics of the initial
conditions, so we are able to present here the cumulants for generic
Non-Gaussian initial conditions, which can be estimated to arbitrary order
including the smoothing effects. The SC model turns out to recover the exact
leading-order non-linear contributions up to terms involving non-local
integrals of the $J$-point functions. We argue that for the hierarchical ratios
$S_J$, these non-local terms are sub-dominant and tend to compensate each
other. The resulting predictions show a non-trivial time evolution that can be
used to discriminate between models of structure formation. We compare these
analytic results to Non-Gaussian N-body simulations, which turn out to be in
very good agreement up to scales where $\sigma \simlt 1$.
