The nonlocal-to-local limit for stochastic conservation laws establishes that as the interaction kernel concentrates to a Dirac delta, solutions of nonlocal stochastic conservation laws converge to solutions of their local counterparts. However, the precise rate at which this convergence occurs, and how that rate depends on the smoothness or roughness of the initial condition, remains unknown. In particular, it is unclear whether initial data with minimal regularity, such as data that is merely a bounded measurable function or an element of some fractional Sobolev space, leads to a degraded convergence rate, or whether the stochastic forcing can in some sense regularize the problem and restore a uniform rate across wide classes of initial data.