I guess that by "isotopy" you mean isotopy of embeddings.
If $B$ is the $n$-sphere and $E=B\times R$ then your question is equivalent to the $(n+1)$-dimensional Schönfliess theorem: known for $n\neq 3$, and a big open question for $n=3$.
To answer your complementary question: if there is a self-diffeomorphism $f$ of $R^{n+1}$ sending the unit sphere $S^n\subset R^{n+1}$ to your embedded sphere $h(S^n)$, then you can always arrange that $f$ is isotopic to the identity (with compact support). Indeed, you can first arrange that $f$ is orientation-preserving; hence $f$ restricted to the unit ball $B^{n+1}$ is an orientation-preserving embedding of the ball; and such an embedding is necessarily isotopic to the inclusion, by the "Alexander trick" (shrinking the ball to its center).