Here is a proof ${\rm Ind}_H^G(V\otimes{\rm Res}_H^GW)\cong({\rm Ind}_H^GV)\otimes W$ are isomorphic using the categorial version of Frobenius reciprocity (that $\rm Ind$, $\rm Res$ are adjoint functors), as well as three other lemmas:
Lemma I. If $\hom_G(A,U)\cong\hom_G(B,U)$ for all $U\in{\rm Rep}(G)$ then $A\cong B$.
Lemma II (Frobenius reciprocity). $\hom_G({\rm Ind}_H^GA,B)\cong\hom_H(A,{\rm Res}_H^GB)$.
Lemma III (currying). $\hom_H(A\otimes B,C)\cong\hom_H(A,B^*\otimes C)$.
Lemma IV. $({\rm Res}_H^GA)^*\cong{\rm Res}_H^G(A^*)$ and $({\rm Res}_H^GA)\otimes({\rm Res}_H^GB)\cong{\rm Res}_H^G(A\otimes B)$.
We employ the last three lemmas to verify the hypothesis of the first lemma: for all $U\in{\rm Rep}(G)$,
$\begin{array}{llll} & \hom_G({\rm Ind}_H^G(V\otimes{\rm Res}_H^GW),U) & \cong & \hom_H(V\otimes{\rm Res}_H^GW,{\rm Res}_H^GU) \\ \cong & \hom_H(V,({\rm Res}_H^GW)^*\otimes{\rm Res}_H^GU) & \cong & \hom_H(V,{\rm Res}_H^G(W^*\otimes U)) \\ \cong & \hom_G({\rm Ind}_H^GV,W^*\otimes U) & \cong & \hom_G(({\rm Ind}_H^GV)\otimes W,U). \end{array}$
Hence ${\rm Ind}_H^G(V\otimes{\rm Res}_H^GW)\cong({\rm Ind}_H^GV)\otimes W.$
In fact the functors ${\rm Ind}_H^G(-\otimes{\rm Res}_H^G-)$ and $({\rm Ind}_H^G-)\otimes-:{\rm Rep}(H)\times{\rm Rep}(G)\to{\rm Rep}(G)$ should be naturally isomorphic (via a canonical isomorphism mt_ gave). All of the isomorphisms in the above calculation are natural, but I am not sure how to successfully upgrade Lemma I to talk about naturality.