I have a collection of real valued random variables on the same probability space indexed by $\mathbb{N}^2$, $\{X_{n,m}\}_{n,m\in\mathbb{N}}$. For each $n \in \mathbb{N}$, I know that $\lim_{m \to \infty} X_{n,m}$ exists and is almost surely finite. Similarly, for each $m \in \mathbb{N}$, $\lim_{n \to \infty} X_{n,m}$ exists and is almost surely finite. Both limits are taken in an almost sure sense.
Normally if I wanted to show that $\lim_{n\to\infty}\lim_{m\to\infty} X_{n,m} = \lim_{m\to\infty}\lim_{n\to\infty} X_{n,m}$, I could do so by showing that $X$ converges uniformly almost surely in either $n$ or $m$ (and apply the Moore-Osgood Theorem). However, I was unable to show uniform convergence in either index. Luckily, I don't actually care too much about the almost sure limit, and I'm only concerned about convergence in distribution (or weak limits). That is, I would like to show that,
$$\text{wlim}_{n\to\infty}\text{wlim}_{m\to\infty} X_{n,m} = \text{wlim}_{m\to\infty}\text{wlim}_{n\to\infty} X_{n,m}.$$
Given that I'm only concerned with convergence in distribution, are there any weaker sufficient conditions under which the interchange of limits holds?
Some more details: We can assume that the support of $\{X_{n,m}\}_{n,m\in\mathbb{N}}$ is compact. I have not been able to identify any monotonicity properties this sequence might hold.
Edit: To be clear, I am asking for sufficient conditions under which the interchange of limits hold. More specifically, I'm hoping for weaker sufficient conditions than if we were to look for an interchange of limits with respect to almost sure convergence. As mentioned by Stephen Montgomery-Smith in the comments, there are simple examples of $\{X_{n,m}\}$ satisfying all the conditions I've already stated for which the interchange of limits fails.
