Let $R_1,R_2$, and $S$ be commutative rings with maps $R_1,R_2 \to S$ and form the fiber product $R = R_1 \times_S R_2$. A well-known theorem of Milnor says that under certain assumptions the category of finitely generated projective $R$-modules is equivalent to a category of "patching data": an object in this category consists of finitely generated projective modules $P_1$ and $P_2$ over $R_1$ and $R_2$ respectively, together with an $S$-isomorphism $P_1 \otimes_{R_1} S \cong P_2 \otimes_{R_2} S$. This may be applied, for instance, to derive a Meyer-Vietoris sequence for Picard groups of rings.

My question: does this generalize to pushouts of schemes? For instance, if I am computing the Picard group of a projective nodal cubic, which can be obtained from $\mathbb{P}^1$ by identifying two points, can I apply a Meyer-Vietoris sequence similar to the one that works in the affine case?