You need to assume projectivity for all primes $p$, otherwise stupid counterexamples are easily found with $G$ the trivial group.

Assuming this, then for any finitely generated $\mathbb Z G$-module $N$, the Ext modules
$Ext^i_{\mathbb Z G}(M,N)$ are finitely generated $\mathbb Z$-modules, whose localisations satisfy
$$Ext^i_{\mathbb Z G}(M,N)\otimes_{\mathbb Z} \mathbb Z_p =
Ext^i_{\mathbb Z_p G}(M \otimes_{\mathbb Z} \mathbb Z_p,N\otimes_{\mathbb Z} \mathbb Z_p)=0.
$$
for $i>0$. To see this, pick a resolution of $M$ by free and finitely generated $\mathbb Z G$-modules and use the flatness of $\mathbb Z_p$ over $\mathbb Z$.

Therefore these Ext modules vanish and $M$ is projective over $\mathbb Z G$.

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