A submanifold $M$ of ${\mathbb R}^{n}$ (with positive codimension) can always (after permuting the coordinates) be locally represented in the form $G_f=\{(x,y) : y=f(x)\}$ where $x$ runs over $V \subset {\mathbb R}^{n-1}$ and $f:V \to {\mathbb R}$. That implies that $M$ is contained in a countable union of such graphs. Each graph has measure zero under the given product measure by Fubini since $A$ is nonatomic. Thus $M$ indeed has measure zero under the product measure by countable additivity.

A submanifold $M$ of ${\mathbb R}^{n}$ (with codimension 1) can always (after permuting the coordinates) be locally represented in the form $G_f=\{(x,y) : y=f(x)\}$ where $x$ runs over $V \subset {\mathbb R}^{n-1}$ and $f:V \to {\mathbb R}$. That implies that $M$ is contained in a countable union of such graphs. Each graph has measure zero under the given product measure by Fubini since $A$ is nonatomic. Thus $M$ indeed has measure zero under the product measure by countable additivity.

A submanifold $M$ of ${\mathbb R}^{n}$ with positive codimension is always a subset of a manifold of codimension 1.