Is there a criteria so that the Tor functor of abelian groups $M$ and $N$ does not vanish?
In particular I'm interested if $\operatorname{Tor}_1 ^\mathbb{Z} (\prod_p \widehat{\mathbb{Z}}_p , \prod_p\widehat{\mathbb{Z}}_p) \neq 0$, where $\widehat{\mathbb{Z}}_p$ denotes the $p$-adic integers.
Thanks in advance.
Over a PID such as $\mathbb{Z}$, any torsion-free module is flat. Thus if either $M$ or $N$ is torsion free, then $\operatorname{Tor}_1^\mathbb{Z}(M,N)=0$. Since the $p$-adic integers are torsion-free, this applies in particular to your example.
More generally, note that $\operatorname{Tor}_1^\mathbb{Z}$ is left-exact in each variable (by the long exact sequence for Tor and the fact that $\operatorname{Tor}_2^\mathbb{Z}$ is always $0$). Also, Tor functors preserve direct limits in either variable, so $\operatorname{Tor}_1^\mathbb{Z}(M,N)$ is the direct limit of $\operatorname{Tor}_1^\mathbb{Z}(A,B)$ where $A$ and $B$ range over all finitely generated subgroups of $M$ and $N$. Since $\operatorname{Tor}_1^\mathbb{Z}$ is left-exact and in particular preserves injections, this means that $\operatorname{Tor}_1^\mathbb{Z}(M,N)=0$ iff $\operatorname{Tor}_1^\mathbb{Z}(A,B)=0$ for all finitely generated subgroups $A$ and $B$ of $M$ and $N$. Now by the classification of finitely generated abelian groups, there exist such $A$ and $B$ with $\operatorname{Tor}_1^\mathbb{Z}(A,B)\neq 0$ iff there is a prime $p$ such that both $M$ and $N$ contain $p$-torsion elements. Thus $\operatorname{Tor}_1^\mathbb{Z}(M,N)\neq0$ iff there is a prime $p$ such that both $M$ and $N$ contain $p$-torsion elements.