I will assume $m=\alpha_d ds$ with $\alpha_d\to \alpha \in [1,\infty)$ independent of $d$. The case $\alpha\to\infty$ is actually easier.

Define $Z=d^{-1} m^{-2} \sum_{i=1}^d \lambda_i^2$. Then $Z$ converges a.s. to $\int x^2 d\mu_\alpha(x)$ where $\mu_\alpha$ is the Pastur-Marchenko distribution of parameter $\lambda=1/\alpha$, see https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution

I will assume $m=\alpha_d d$ with $\alpha_d\to \alpha \in [1,\infty)$ independent of $d$. The case $\alpha\to\infty$ is actually easier.

Define $Z=d^{-1} m^{-2} \sum_{i=1}^d \lambda_i^2$. Then $Z$ converges a.s. to $\int x^2 d\mu_\alpha(x)$ where $\mu_\alpha$ is the Pastur-Marchenko distribution of parameter $\lambda=1/\alpha$, see https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution