The consistency is proved by Jeff Cheeger, M\"uller and Schrader in 1984, "On the Curvature of Piecewise Flat Sapces". Roughly speaking, given a smooth Riemannian manifold with a smooth metric, there exists a sequence of triangulation, on which Regge's definition converges to the smooth curvature as a measure.
At the linearized level, there is also a recent paper on the consistency: Christiansen 2011, "On The Linearization of Regge Calculus". One of the theorems in the paper is that we have consistency between linearized Regge and linearized Einstein equation as well.
That said, when you talk about the convergence of a numerical algorithm, it depends on a lot of other things as well, such as your formulation of the Einstein's equation. You will also need some form of stability to ensure convergence. Those questions remain to be solved (hopefully in my thesis:-).