As a first step, we reduce the assertion to the case where $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ is $\pi$-ample. Indeed, the set of vertical $\pi$-ample $\mathbf{R}$-divisors, which is an open convex cone in $\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{R}}$, is non-empty by (i) of Lemma 1.4. We may thus choose a basis $A_{1},...,A_{r}$ of $\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{R}}$ made up of $\pi$-ample Cartier divisors. Let $\varepsilon=(\varepsilon_{i})\in\mathbf{R}_{+}^{r}$ be such that $D_{\varepsilon}:=D+\sum_{i}\varepsilon_{i}A_{i}$ is a $\mathbf{Q}$-divisor. The fact that $D$ is $\pi$-nef means that $D\cdot C\geq 0$ for each curve $C$ contained in a fiber of $\pi$. Since each $A_{i}$ is $\pi$-ample, it follows from Kleiman’s criterion [Kle66] that $D_{\varepsilon}$ is $\pi$-ample on the projective $k$-scheme $\mathcal{X}_{0}^{\prime}$, hence $D_{\varepsilon}$ is also $\pi$-ample on $\mathcal{X}^{\prime}$ by [EGA, III.4.7.1]. Upon replacing $D$ with $D_{\varepsilon}$ for $\varepsilon$ arbitrarily small we may thus assume as desired that $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ is $\pi$-ample.

Now choose $m\gg 1$ such that $\mathcal{O}_{\mathcal{X}^{\prime}}(mD)$ is $\pi$-globally generated, which means that the vertical fractional ideal sheaf $\mathfrak{a}:=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}(mD)$ satisfies $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(mD)$. It is obvious that $\mathfrak{a}\subset\mathcal{O}_{\mathcal{X}}(m\pi_{*}D)$, hence $\mathcal{O}_{\mathcal{X}^{\prime}}(mD)\subset\mathcal{O}_{\mathcal{X}^{\prime}}(m\pi^{*}\pi_{*}D)$, and the result follows. ∎