Useful, thanks. I'll leave it open for a bit to see if perhaps I can get anything tighter. Seems unlikely I guess; I know there are some nonzero off diagonal elements in $\Sigma$, but I don't know where or how large they are.

Vague by intention :) I was sort of fishing. Initially I had taken $V$ diagonal (wlog up to a rotation of $x$). Here we arrive at the same place as @Mark Meckes generalized to any $V$. That is, taking $||x||_V^2 = x'V^{-1}x$ then $||x||_V$ and $x/||x||_V$ are independent when $x\sim N(0, V)$. Should probably dust off my linear models text!

Yeah, the $v_i$'s should have been inverted. Fixed now. As you correctly inferred, I was using $||x|| = \sqrt(x'x)$, I made them into $||x||_2$ which is hopefully clearer. Thanks