If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms
$$\mathcal N\_p (\varphi)= \sum\_{|\alpha|, |\beta| \leq p} \sup\_{x\in \mathbb R^n} |x^\alpha \partial^\beta \varphi (x) |\, ,$$
Is it true that $\mathcal S(\mathbb R^n) \hat \otimes\_\pi \mathcal S(\mathbb R^m)\simeq \mathcal S(\mathbb R^{n+m})$, where $\mathcal S(\mathbb R^n) \hat \otimes\_\pi \mathcal S(\mathbb R^m)$ is the completion of $\mathcal S(\mathbb R^n) \otimes \mathcal S(\mathbb R^m)$ for the topology $\pi$ defined by the family of semi-norms 
$$\mathcal N\_{p,q} (\varphi\otimes \psi)=\mathcal N\_p(\varphi)\mathcal N\_q (\psi)\, .$$



$\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?
Yes, if you understand the tensor product topology in the right way. 
Since the spaces are nuclear, inductive and projective tensor products coincide.
The result is theorem 51.6 of Treves: Topological vector spaces, distributions, and kernels.


Added later:
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Attention: The description of seminorms on the tensor-product given in the question is not sufficient to specify a locally convex topology on the tensor product. There are many satisfying this description; all between the the projective one and the the inductive one and even more. See the source I have given, or many other books.


