We have

$$c_{1}\left(\overline{\mathcal{O}}^{h\circ p_{\mathfrak{X}^{\prime}}}\right)^{n}=\Deg(S)\cdot n!\cdot\MA(h)$$

on $p_{\mathfrak{X}^{\prime}}^{-1}(\tau)$, where $\MA(h)$ is understood to be a measure on $\mathfrak{X}^{\prime\textup{an}}$ by pushforward with the inclusion $\tau\hookrightarrow\mathfrak{X}^{\prime\textup{an}}$.