A nice version of the manifold version of Fubini's theorem is in Differentialgeometrie und Fasserbündel by Rolf Sulanke and Peter Wintgen:

Let $\phi \in C^1(M,N)$, where $M,N$ are smooth manifolds of dimensions $m,n$, respectively, with $m \ge n$. Let $\omega \in \Omega^{m-n}(M)$ and $\eta \in \Omega^n(N)$, and let $f : M \to {\bf R}$ be measurable (meaning, its superposition with any map is Lebesgue measurable). Assume the set of critical values of $\phi$ has measure zero in $N$ (again, this means the image under any map has Lebesgue measure zero).

If the $m$-form $f\omega\wedge\phi^*\eta$ is integrable on $M$, then for almost all $x \in N$ the integral:

$$ \int\limits_{\phi^{-1}(x)} f\omega $$

...is well-defined and, moreover, when treated as a function of $x$ and multiplied by $\eta$, it's integrable on $N$ and:

$$ \int\limits_M f\omega\wedge\phi^*\eta = \int\limits_N \bigg( \int\limits_{\phi^{-1}(x)} f\omega \bigg)\, \eta $$

A nice version of the manifold version of Fubini's theorem is in Differentialgeometrie und Fasserbündel by Rolf Sulanke and Peter Wintgen:

Let $\phi \in C^1(M,N)$, where $M,N$ are smooth manifolds of dimensions $m,n$, respectively, with $m \ge n$. Let $\omega \in \Omega^{m-n}(M)$ and $\eta \in \Omega^n(N)$, and let $f : M \to {\bf R}$ be measurable (meaning, its superposition with any map is Lebesgue measurable). Assume the set of critical values of $\phi$ has measure zero in $N$ (again, this means the image under any map has Lebesgue measure zero).

If the $m$-form $f\omega\wedge\phi^*\eta$ is integrable on $M$, then for almost all $x \in N$ the integral:

$$ \int\limits_{\phi^{-1}(x)} f\omega $$

is well-defined and, moreover, when treated as a function of $x$ and multiplied by $\eta$, it is integrable on $N$ and:

$$ \int\limits_M f\omega\wedge\phi^*\eta = \int\limits_N \bigg( \int\limits_{\phi^{-1}(x)} f\omega \bigg)\, \eta $$