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 a regular variety over a not necessarily algebraically closed field.

 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.

*Edit:* As the OP remarks in his comments below, the formula 
$Pic(\mathbb P(\mathcal E) )=p^\ast Pic(Y)\oplus \mathbb Z \xi $
is also true  for any integral variety $Y$ over an algebraically closed field, locally factorial or not. The tool is  then   Grauert's semi-continuity theorem (cf. Hartshorne Chapter III, §12) rather than Chow groups.