Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$ when $Y$ is regular.
If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have
$$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n-1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $ (This is true even if $Y$ is not regular)
If you remember that locally factorial varieties (for example regular or smooth varieties) satisfy $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis of local factoriality.
If $Y$ has bad singularities, I wouldn't bet on the correctness of the formula since there exist (non semi-normal) varieties $Y$ such that the canonical morphism $\pi^\ast: Pic (Y) \to Pic (Y) \times \mathbb A^1 $ is not surjective.