Lets define n arbitrarily scaled, rotated, located cuboids, and plane by normalized direction vector.
Cuboids is supposed to be "flattened" by orthogonal projection on the plane, so we can treat them like a 2d figures.
The task is to find an algorithmic effective method to calculate the total area (inaccuracy is fine too) occupied by figures on that 2d space, bypassing doubling due to overlaps.
It is difficult to exploit the fact that you are projecting cuboids (rather than more complex objects), and difficult to exploit the randomness of the projection direction. So likely the best algorithm is just a generic plane-sweep algorithm (also known as "line-sweep"). You can find notes for these algorithms all over the web, usually computing the intersection of objects but only minor modifications are needed to construct the union.
Since you mention "inaccuracy," there has been attention to inaccurate, "toleranced polygons":
Cazals, Frédéric, and G. D. Ramkumar. "Algorithms for computing intersection and union of toleranced polygons with applications." AI EDAM 11, no. 4 (1997): 263-272.
If you care to delve into the combinatorial complexity issues, start here:
Pach, János, Pankaj K. Agarwal, and Micha Sharir. State of the union (of geometric objects). American Mathematical Society, 2008.
There you will find this nice theorem for a restricted class of shapes: The complexity of the union is linear rather than quadratic.
THEOREM 2.2 (Kedem et al.). Let $C = \{C_1, C_2, . . . , C_n\}$ be a family of $n \ge 3$ pseudo-disks in the plane. Then the boundary of $U(C)$ consists of at most $6n − 12$ elementary arcs, and this bound is tight in the worst case.
The boundaries of any two pseudo-disks cross at most twice.
