I think a relevant reference which is worthwhile to have a look is:
Wall, C. T. C. Classification of (n−1)-connected 2n-manifolds. Ann. of Math. (2) 75 1962 163–189.
Let me quote one result from this paper:
Theorem 5. If $\pi_{n-1}(SO)=0$ ,$n\geq 3$,and $M_1$ and $M_2$ are differential $(n-1)$ connected $2n$-manifolds of the same homotopy type,then for some manifold $T$ homeomorphic (and so combinatorially equivalent) to $S^{2n}$, $M_1$ is diffeomorphic to $M_2\sharp T$. If $n=3,6$, $M_1$ is diffeomorphic to $M_2$.
This tells you that there is no differentiable fake $S^n\times S^n$ for $n\equiv 6\mod8$