We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of coin tossing Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of $1/4$ with respect to every $p$-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than $1/4$. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points.