We compute the Szlenk index of an arbitrary projective tensor product $C(K)\widehat{\otimes}_πC(L)$ 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 $C(K)\widehat{\otimes}_πC(L)$ and $C(M)$ or $C(M)\widehat{\otimes}_πC(N)$ for particular choices of $K,L,M,N$.