For $p=1$ one can bound the 1-Wasserstein metric by 
$$|m-n| + \sqrt{\sum_{i=1}^{d} \left[ 
\left(
\sqrt{\lambda_i} - \sqrt{\gamma_i}\right)^2 + 2\sqrt{\lambda_i\gamma_i}(1-v_i)\cdot u_i
\right]}$$

when $\lambda_i$ and $\gamma_i$ are the $i^{th}$ eigen values of $U$ and $V$ respectively, $v_1,\ldots,v_d$ and $u_1,\ldots,u_d$  are the corresponding orthonormal basis of eigen-vectors.

See [Chafai & Malrieu](https://arxiv.org/pdf/0805.0987.pdf) Lemma 2.4. 

Although this bound seems close-in nature to the $p=2$ bound, I'm not sure if it is sharp.