  We investigate all geodesics in the entire class of nonexpanding impulsive
gravitational waves propagating in an (anti-)de Sitter universe using the
distributional metric. We extend the regularization approach of part I,
[SSLP16] to a full nonlinear distributional analysis within the geometric
theory of generalized functions. We prove global existence and uniqueness of
geodesics that cross the impulsive wave and hence geodesic completeness in full
generality for this class of low regularity spacetimes. This, in particular,
prepares the ground for a mathematically rigorous account on the 'physical
equivalence' of the continuous with the distributional `from' of the metric.
