This seems to be a typical case where one can apply Prokhorov's theorem.

Since both sequences $(Z^N)$ and $(W^N)$ converge in distribution, both families of distributions are tight due to Prokhorov theorem. It easily follows that the sequence of couples $(X^N,Z^N)$ is tight, and again due to Prokhorov theorem, it is relatively compact, and we have only to see that there is a unique limiting point in distribution for any subsequence of $(X^N,Z^N)$. But each limiting point has to have the characteristic functional that you give in the r.h.s., and this characterizes the limiting distribution uniquely.

This seems to be a typical case where one can apply Prokhorov's theorem.

Since both sequences $(Z^N)$ and $(W^N)$ converge in distribution, both families of distributions are tight due to Prokhorov theorem. It easily follows that the sequence of couples $(Z^N,W^N)$ is tight, and again due to Prokhorov theorem, it is relatively compact, and we have only to see that there is a unique limiting point in distribution for any subsequence of $(Z^N,W^N)$. But each limiting point has to have the characteristic functional that you give in the r.h.s., and this characterizes the limiting distribution uniquely.