I'll leave this here in case anyone can find my error.
Assume the same setup as in @RobertIsrael's response. As he showed, we have that $\textrm{Cov}\left[u,v\right]=\textrm{E}\left[uv\right]=c^2s_x^2-d^2s_y^2=0$. Now you know something about $s_x$ and $s_y$, but you still can't say anything about $s_u$ and $s_v$, so let's try to find out more about them. If $u$ and $v$ have mean zero, then their variance can be written as $\textrm{Var}\left[u\right]=\textrm{E}\left[u^2\right]=s_u^2$ and $\textrm{Var}\left[v\right]=\textrm{E}\left[v^2\right]=s_v^2$.
$\begin{align} s_u^2 &= \textrm{E}\left[\left(cx+dy\right)^2\right]\\ &= \textrm{E}\left[c^2x^2+2cdxy+d^2y^2\right]\\ &= c^2s_x^2+2cd\textrm{Cov}\left[x,y\right]+d^2s_y^2\\ &= c^2s_x^2+2cds_xs_yR+d^2s_y^2\\ \end{align}$
$\begin{align} s_v^2 &= \textrm{E}\left[\left(cx-dy\right)^2\right]\\ &= \textrm{E}\left[c^2x^2-2cdxy+d^2y^2\right]\\ &= c^2s_x^2-2cd\textrm{Cov}\left[x,y\right]+d^2s_y^2\\ &= c^2s_x^2-2cds_xs_yR+d^2s_y^2\\ \end{align}$
$\begin{align} s_us_v &= \sqrt{\left(c^2s_x^2+2cds_xs_yR+d^2s_y^2\right)\left(c^2s_x^2-2cds_xs_yR+d^2s_y^2\right)}\\ &= \sqrt{c^4s_x^4-4c^2d^2s_x^2s_y^2R^2+2c^2d^2s_x^2s_y^2+d^4s_y^4}\\ &= \sqrt{\left(c^2s_x^2-d^2s_y^2\right)^2+4c^2d^2s_x^2s_y^2\left(1-R^2\right)}\\ &= \sqrt{4c^2d^2s_x^2s_y^2\left(1-R^2\right)}\\ &= 2cds_xs_y\sqrt{1-R^2}\\ \end{align}$