Note that \[ P(x) = x + 2x^2 + 3x^3 + \cdots + 24x^{24} + 23x^{25} + 22x^{26} + \cdots + 2x^{46} + x^{47}, \]and \[ xP(x) = x^2 + 2x^3 + 3x^4 + \cdots + 24x^{25} + 23x^{26} + \cdots + 2x^{47} + x^{48}, \]so \begin{align*} (1-x)P(x) &= x+x^2+\cdots + x^{24} - (x^{25} + x^{26} + \cdots +x^{47} + x^{48}) \\ &=(1-x^{24})(x+x^2+\cdots +x^{24}).\end{align*}Then, for $ x\ne1 $, \begin{align*} P(x) &={{x^{24}-1}\over{x-1}} \cdot x(1+x+\cdots +x^{23})\\ &=x\Bigl({{x^{24}-1}\over{x-1}}\Bigr)^2\; .&(*) \end{align*}One zero of $ P(x) $ is 0, which does not contribute to the requested sum. The remaining zeros of $ P(x) $ are the same as those of $ (x^{24}-1)^2 $, excluding 1. Because $ (x^{24}-1)^2 $ and $ x^{24}-1 $ have the same distinct zeros, the remaining zeros of $ P(x) $ can be expressed as $ z_k= {\rm cis}\,15k^{\circ} $ for $ k = 1,2,3,\dots,23 $. The squares of the zeros are therefore of the form $ {\rm cis}\,30k^{\circ} $, and the requested sum is $$\sum_{k=1}^{23}|\sin30k^{\circ}|= 4\sum_{k=1}^{5}|\sin30k^{\circ}| =4\left( 2 \cdot \frac{1}{2} + 2 \cdot \frac{\sqrt{3}}{2} + 1 \right) = \boxed{8+4\sqrt3}.$$Note: The expression $ (*) $ can also be obtained using the identity $$(1+x+x^2+\cdots +x^{n})^2 = 1+2x+3x^2+\cdots+(n+1)x^{n}+\cdots+3x^{2n-2}+2x^{2n-1}+x^{2n}.$$