Nonholonomic integrators are a class of geometric numerical integration schemes that are designed to simulate mechanical systems with nonholonomic constraints. To the best of our knowledge, so far there have been no variational integrators designed for stochastic systems with noisy nonholonomic constraints, which are extensively studied in robotics and control area. Based on the stochastic nonholonomic variational formulation introduced in Part I, we present a stochastic integrator for both stochastically unconstrained and stochastically nonholonomic systems under the same framework. The numerical integration scheme is obtained by deriving a discrete counterpart of the stochastic variational principle discussed in Part I.