= Solution
For $u=e^{2\pi i y}$ and $w=e^{2\pi i z}$, define
$$
\Phi:M\longrightarrow N,
\qquad
\Phi([t,(u,w)])=\Gamma v(-t,y,z).
$$
Changing $y$ or $z$ by an integer changes $v(-t,y,z)$ by left multiplication by an element of $\Gamma$, so $\Phi$ is independent of the chosen lifts. It also respects the mapping-torus seam because
$$
v(1,0,0)v(-1,y,z)=v(0,y+z,z),
$$
which is exactly the identification
$$
(1,(u,w))\sim(0,(uw,w))=(0,f(u,w)).
$$
Every $\Gamma$-orbit has a representative with its first coordinate in $[-1,0]$, so $\Phi$ is surjective. If two such representatives lie in one orbit, the integer shift of their first coordinates is zero unless they are opposite endpoints; in the first case their torus coordinates agree, and in the endpoint case they are related by the displayed seam. Thus $\Phi$ is injective. It is continuous, $M$ is compact, and $N$ is Hausdorff, so a continuous-bijection theorem gives
$$
\boxed{\ M\cong N.\ }
$$
This realizes the <Heisenberg nilmanifold as a torus mapping torus>.
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