Let me work out the case $\mu_i=\mu$, $\sigma_i^2=\sigma$, which is easiest to formulate. For large $k$ the variable $Y^2$ has a Gaussian distribution with mean $M=k[(\mu/\sigma)^2+1]$ and variance $V=4k[(\mu/\sigma)^2+1/2]$. It follows that, to leading order in $1/k$, $$\mathbb{E}[Y] =\int_{-\infty}^\infty dz\, (2\pi V)^{-1/2}e^{-z^2/2V}(\sqrt{M}+\tfrac{1}{2}zM^{-1/2}-\tfrac{1}{8}z^2M^{-3/2})=M^{1/2}-\tfrac{1}{8}VM^{-3/2}$$ $$=k^{1/2}\sqrt{(\mu/\sigma)^2+1}+{\cal O}(k^{-1/2}),$$ $$\text{var}\,[Y]=M-\mathbb{E}[Y]^2=\frac{(\mu/\sigma)^2+1/2}{(\mu/\sigma)^2+1}+{\cal O}(k^{-1}).$$ So the mean of $Y$ scales as $\sqrt{k}$ while the variance tends to a constant in the limit $k\rightarrow\infty$.
As a check, I plot the mean and variance of $Y$ as a function of $k$ for $\mu=1=\sigma^2$: exact = gold, asymptotic = blue:
