  We compute the Szlenk index of an arbitrary projective tensor product
$C(K)\widehat{\otimes}_\pi C(L)$ of spaces $C(K), C(L)$ of continuous functions
on scattered, compact, Hausdorff spaces. In particular, we show that it is
simply equal to the maximum of the Szlenk indices of the spaces $C(K), C(L)$.
We deduce several results regarding non-isomorphism of
$C(K)\widehat{\otimes}_\pi C(L)$ and $C(M)$ or $C(M)\widehat{\otimes}_\pi C(N)$
for particular choices of $K,L,M,N$.
