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(For the record, I am summarizing the comments here.) Deligne and Fulton have shown that fundamental group of the complement of a nodal curve in $\mathbb{C}\mathbb{P}^2$ is abelian. It follows easily that in the case of smooth curve $C$ of degree $d$, the fundamental group is $\mathbb{Z}/d$. Therefore the universal cover in this case is the complement of the branch curve in the $d$-sheeted cyclic cover of the plane branched over $C$. See above comments for further historical remarks.