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.