Let $\overline{K}/\overline{F}$ be algebraic closures. The groups of components of the Picard groups $\Pic(Y_{\overline{F}})$ and $\Pic(Y_{\overline{K}})$ are then isomorphic, so we have an isomorphism $N^{1}(Y_{\overline{F}})_{\mathbf{Q}}\simeq N^{1}(Y_{\overline{K}})_{\mathbf{Q}}$ by the result of [Mat57] recalled above (compare [MP11, Proposition 3.1]). This isomorphism is furthermore compatible with ample classes by the Nakai-Moishezon criterion for ampleness.

It is enough to show the surjectivity of $N^{1}(Y)_{\mathbf{Q}}\to N^{1}\left(Y_{K}\right)_{\mathbf{Q}}$. Let $\beta\in N^{1}(Y_{K})_{\mathbf{Q}}$. By the previous result, we find $L\in\Pic(Y_{\overline{F}})_{\mathbf{Q}}$ mapping to the lift of $\beta$ in $N^{1}(Y_{\overline{K}})_{\mathbf{Q}}$. Since $Y_{\overline{F}}$ is in particular reduced, $L$ can be represented by some $\bar{D}\in\Div(Y_{\overline{F}})_{\mathbf{Q}}$. The average of the Galois orbit of $\bar{D}$ is then $\mathrm{Gal}(\bar{F}/F)$-invariant, hence descends to $D\in\Div(Y)_{\mathbf{Q}}$ by [Car58, Proposition 11, §4.6]. By construction, the image of $D$ under the composition

coincides with the image of $\beta$. But $N^{1}(Y_{K})_{\mathbf{Q}}\to N^{1}(Y_{\overline{K}})_{\mathbf{Q}}$ is injective by the projection formula, and the result follows. ∎