There are a few references worth mentioning:

• https://hal.archives-ouvertes.fr/hal-00915365/document
• https://www.lpsm.paris/pageperso/bolley/bgv.pdf
• https://arxiv.org/pdf/1804.10556.pdf

Example of a bound you could hope to get

Let $d \ge 2p$ be the dimensionality of your square. Suppse $\mu$ is light-tailed, i.e suppose there exists $\alpha > p$ such that $\int \exp(|X|^\alpha) d\mu < \infty $. Then for every $t > 0$, one has $$ P(W_p(\mu,\hat{\mu}_n) \ge t) \le \begin{cases}c_1\exp(-c_2Nt^{d/p}),&\mbox{ if }t \le 1,\\ c_1\exp(-c_2Nt^{\alpha/p}),&\mbox{ else,}\end{cases} $$ where $c_1$ and $c_2$ are positive constants independent of $N$ and $t$. This is a direct consequence of Theorem 2 of the first paper.

You can get an even finer bound by using Proposition 10.