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No. Take $X$ to be the union of two smooth, genus $0$ curves, say $X_1$ and $X_2$, that intersect in two nodes, say $p$ and $q$. Then $\text{Pic}(X)$ is an extension of $\mathbb{Z}^2$ by $\mathbb{G}_m$. So let $Y$ be a geometric $\mathbb{A}^1$-bundle over $X$ whose restriction to each of $X_1$ and $X_2$ is trivial, yet that is nontrivial on $X$. Identify $X$ with the zero section of this $\mathbb{A}^1$-bundle. The normal cone is a normal bundle, essentially just the data of $Y$ once again. Yet you cannot recover $Y$ from its restriction to the two irreducible components.