EDIT: I cannot delete this post as it's been already accepted :(
The proof is carried out by induction on $m.$
$m=1$ is trivial by choosing coordinates $(x^{(1)}_0,x^{(1)}_1,...,x^{(1)}_{n_1})$ where $\sum_{j=0}^{n_1} (x_j^{(1)})^2=1$, so let $m=2,$ then similarly $S^{n_1} \times S^{n_2}$ is embedded in $\mathbb{R}^{n_1+n_2+2}$ which also lies in the hypersurface $H$ with equation $\sum_{j=0}^{n_1} (x_j^{(1)})^2+\sum_{j=0}^{n_2} (x_j^{(2)})^2=2.$ In fact, $H$ is diffeomorphic to $S^{n_1+n_2+1}.$ Now, the embedding is missing at least a point, for example $p=(0,...,0,1,1) \in H,$ so by stereographic projection $S^{n_1+n_2+1} \setminus \{p\}$ is diffeomorphic to $\mathbb{R}^{n_1+n_2+1}.$
Suppose that the assertion holds for $