Citing Wikipedia:
Let $U\subseteq \mathbb {R} ^{m}$ and $V\subseteq \mathbb {R} ^{n}$ be open sets. Assume all vector spaces to be over the field $\mathbb {F}$, where $\mathbb{F} =\mathbb {R}$ or $\mathbb {C}$. For $f\in {\mathcal {D}}(U\times V)$ define for every $u \in U$ and every $v\in V$ the following functions: $$ {\begin{alignedat}{9}f_{u}:\,&V&&\to \,&&\mathbb {F} &&\quad {\text{ and }}\quad &&f^{v}:\,&&U&&\to \,&&\mathbb {F} \\&y&&\mapsto \,&&f(u,y)&&&&&&x&&\mapsto \,&&f(x,v)\\\end{alignedat}}$$ Given $S\in {\mathcal {D}}^{\prime }(U)$ and $T\in {\mathcal {D}}^{\prime }(V)$, define the following functions: $${\begin{alignedat}{9}\langle S,f^{\bullet }\rangle :\,&V&&\to \,&&\mathbb {F} &&\quad {\text{ and }}\quad &&\langle T,f_{\bullet }\rangle :\,&&U&&\to \,&&\mathbb {F} \\&v&&\mapsto \,&&\langle S,f^{v}\rangle &&&&&&u&&\mapsto \,&&\langle T,f_{u}\rangle \\\end{alignedat}}$$ where $\langle T,f_{\bullet }\rangle \in {\mathcal {D}}(U)$ and $\langle S,f^{\bullet }\rangle \in {\mathcal {D}}(V)$. These definitions associate every $S\in {\mathcal {D}}'(U)$ and $T\in {\mathcal {D}}'(V)$ with the (respective) continuous linear map: $${\begin{alignedat}{9}\,&{\mathcal {D}}(U\times V)&&\to \,&&{\mathcal {D}}(V)&&\quad {\text{ and }}\quad &&\,&&{\mathcal {D}}(U\times V)&&\to \,&&{\mathcal {D}}(U)\\&f&&\mapsto \,&&\langle S,f^{\bullet }\rangle &&&&&&f&&\mapsto \,&&\langle T,f_{\bullet }\rangle \\\end{alignedat}}$$ Moreover if either $S$ (resp. $T$) has compact support then it also induces a continuous linear map of $C^{\infty }(U\times V)\to C^{\infty }(V)$ (resp. $C^{\infty }(U\times V)\to C^{\infty }(U)$).
Fubini's theorem for distributions. Let ${\displaystyle S\in {\mathcal {D}}'(U)}$ and ${\displaystyle T\in {\mathcal {D}}'(V)}$. If ${\displaystyle f\in {\mathcal {D}}(U\times V)}$ then $\langle S,\langle T,f_{\bullet }\rangle \rangle =\langle T,\langle S,f^{\bullet }\rangle \rangle.$ The tensor product of ${\displaystyle S\in {\mathcal {D}}'(U)}$ and $T\in {\mathcal {D}}'(V)$, denoted by $S\otimes T$ or $T\otimes S$ is the distribution in $U\times V$ defined by: $${\displaystyle (S\otimes T)(f):=\langle S,\langle T,f_{\bullet }\rangle \rangle =\langle T,\langle S,f^{\bullet }\rangle \rangle}$$
Question. Does that mean the tensor product is commutative? It's not true in general, though
No, the tensor product of distributions is not commutative.
If we rewrite the last line with formal arguments we have $$ \langle (S \otimes T)(x,y), f(x,y) \rangle = \langle S(x), \langle T(y), f(x,y) \rangle \rangle = \langle T(y), \langle S(x), f(x,y) \rangle \rangle . $$ Had the tensor product been commutative then the last expression would have read $\langle T(x), \langle S(y), f(x,y) \rangle \rangle$ or $\langle T(y), \langle S(x), f(y,x) \rangle \rangle$.