This is false; the differentials do not necessarily have image zero.
The problem comes earlier when you want to define the "reduced" groups: the map $H^p_G(M;\Bbb Z) \to H^p_G(\ast;\Bbb Z)$ is not necessarily surjective.
A standard example would be the case where $G = C_2$ is the cyclic group of order two and $M = S^n$ is the n-sphere with the antipodal action of $G$. In that case, $EG \times_\phi M$ is homotopy equivalent to the real projective space $\Bbb{RP}^n$, and in particular its integer cohomology vanishes in degrees greater than $n$. However, $$E_2^{p,0} = H^p_{C_2}(\ast;\Bbb Z) \cong H^p(BC_2;\Bbb Z)$$ is nontrivial for all even values of $p$.