Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively.
Then there is an induced principal $\Spin_{n+n'}$ bundle over the product $M\times M'$. Is there a standard notation or symbol used to denote this bundle?
Also,
Is there a notation for the associated $\mathbb R^{n+n'}$ bundle over $M \times M'$, in terms of the associated $\mathbb R^n$ bundle on $M$ and the associated $\mathbb R^{n'}$ bundle on $M'$?
This is not particular to $\Spin$ at all; one could ask this question of any group $H$ containing a direct product $K \times K'$ inside it, and constructing an $H$ bundle over $M \times M'$ given a $K$ bundle over $M$ and a $K'$ bundle over $M'$.
In case there's any ambiguity: The bundle structure is induced by the homomorphism $$ \Spin_n \times \Spin_{n'} \to \Spin_{n+n'} $$ which lives over the homomorphism $$ O(n) \times O(n') \to O(n+n') $$ given by sending matrices $A_n$ and $A_{n'}$ to a block diagonal matrix with entries $A_n$ and $A_{n'}$. Then the induced transition maps are defined by $$ g_{U \times U', V \times V'}: (x,x') \mapsto (\phi_{UV}(x),\phi'_{U'V'}(x')) \in \Spin_n \times \Spin_{n'} \subset \Spin_{n+n'}. $$ where $\phi$ and $\phi'$ are the transition maps of the bundles $P$ and $P'$, respectively.