There is indeed an example as requested in the question.$\newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}}$
We know that if $M$ and $N$ are two h-cobordant closed topological manifolds (of dimension greater than 4), then $M\times\RR$ is homeomorphic to $N\times\RR$ (for example, see Oscar Randal-Williams' comment here).
So it suffices to find closed manifolds $M$ and $N$ which are h-cobordant and not homeomorphic, and moreover $M$ is null-cobordant; then $N$ is also null-cobordant. An example of such manifolds is given by Farrel and Hsiang in their article $H$-cobordant manifolds are not necessarily homeomorphic: we can take $M=L\times\TT^{n-3}$, where $L$ is a certain 3-dimensional lens space, and $\TT^{n-3}$ is a torus of dimension at least 3. Note that $M$ is null-cobordant since $\TT^{n-3}$ is null-cobordant.