If both probability measures have smooth densities, the total variation can be bounded by Wasserstein (even by a weaker metric Levy-Prokhorov), see [this][1] paper. 

More precisely, see Lemma 5.1 and proof of Theorem 2.1. The basic idea is given below.

If both p and q are smooth densities, d_V(p, p_gamma) and d_V(q, q_gamma) are sufficiently small for every small gamma, where d_V is the total variation and p_gamma is a convolution of p and the uniform density as defined in the paper.

Note that d_V(p, q) < d_V(p, p_gamma) + d_V(p_gamma, q_gamma) + d_v(q_gamma, q) by the triangle inequality.

Lemma 5.1 guarantees that if p and q are close in Levy-Prokhorov metric, then d_V(p_gamma, q_gamma) is also small provided that gamma is much bigger than the Levy-Prochorov distance.


  [1]: https://projecteuclid.org/euclid.ejs/1511492460