By exam $|\wh{G\times H}|=|\wh G||\wh H|$. Thus it suffices to prove that 1. all representations $(g,h)\mapsto\Psi(g)\otimes\Phi(h)$ in $E\otimes F$ are irreducible, provided $\Psi$ and $\Phi$ are irreducible. 2. all these representations are pairwise inequivalent. As for 1. the trace of this representation is $\psi\otimes\vp$ and its norm is $$ \Vert\psi\otimes\vp\Vert_{L_2(G\times H)} =\Vert\psi\Vert_{L_2(G)}\Vert\vp\Vert_{L_2(H)} =1 $$ i.e. $\psi\otimes\vp$ is a character and thus $\Psi\otimes\Phi$ is irreducible. 2. If two such representations $\Psi_j\otimes\Phi_j$, $j=1,2$, are equivalent, then their characters coincide, which can only happen if both $\psi_1=\psi_2$ and $\vp_1=\vp_2$.
Can you find an argument proving the irreducibility of $(g,h)\mapsto\Psi(g)\otimes\Phi(h)$ directly, i.e. without refering to character theory?