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Building on my comment just a bit, here is a general upper/lower bound on the Wasserstein distance between $\mu$ and $\nu$ based on a maximal coupling with respect to their total variation distance: $$ (1/2) \min_{x_i \ne x_j} c(x_i, x_j) \sum_{k=1}^n |1/n - \nu_k| \le \mathcal{W}(\mu,\nu) \le (1/2) \max_{x_i \ne x_j} c(x_i, x_j) \sum_{k=1}^n |1/n - \nu_k| \;. $$