You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then
$$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$
This can be further reduced to
$$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$
with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply
$$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$
so $Z$ has for large $M$ a chi-squared distribution with 2 degrees of freedom.