The paper that proved the linear $(6n-12)$ complexity of the boundary of the union of $n$ Jordan
curves that intersect pairwise at most twice,
K. Kedem, R. Livne, J. Pach, and M. Sharir, "On the union of Jordan regions and collision-free
translational motion amidst polygonal obstacles," Discrete Comput. Geom., 1 (1986), 59–71.
also showed that one can construct this boundary in $O(n \log^2 n)$ time. See Section 4, "Efficient Calculation of $K$," 66ff. The algorithm is divide-and-conquer, and employs a plane-sweep idea
of Ottmann, Widmeyer, and Wood.
Note that the result neither requires unit radius nor even circles, but only Jordan curves that intersect pairwise at most twice, which of course includes circles (and so unit circles).
Here is the first two sentences of the abstract of the above paper:
And here is a figure illustrating the proof for disks in the later survey paper,
"State of the Union (of Geometric Objects),"
by Agarwal, Pach, Sharir (Proc. Joint Summer Research Conference on Discrete and Computational Geometry--Twenty Years later, Contemp. Math
, AMS, Providence, RI, 154):