In this second portion of a two-part analysis of a scalable computa- tional approach to stochastic unit commitment, we focus on solving stochastic mixed-integer programs in tractable run-times. Our solution technique is based on Rockafellar and Wets' progressive hedging algorithm, a scenario-based decomposi- tion strategy for solving stochastic programs. To achieve high-quality solutions in tractable run-times, we describe critical, novel customizations of the progressive hedging algorithm for stochastic unit commitment. Using a variant of the WECC- 240 test case with 85 thermal generation units, we demonstrate the ability of our approach to solve realistic, moderate-scale stochastic unit commitment problems with reasonable numbers of scenarios in no more than 15 minutes of wall clock time on commodity compute platforms. Further, we demonstrate that the result- ing solutions are high-quality, with costs typically within 1-2.5% of optimal. For larger test cases with 170 and 340 thermal generators, we are able to obtain solu- tions of identical quality in no more than 25 minutes of wall clock time. A major component of our contribution is the public release of the optimization model, as- sociated test cases, and algorithm results, in order to establish a rigorous baseline for both solution quality and run times of stochastic unit commitment solvers.