I think that there is no formula. The best one can do is to estimate. Here is a simpler problem of the same sort: suppose you have a parametrization of the boundary of a simply connected region, and suppose that 0 is inside. Consider the Riemann mapping f of this region sending 0 to 0. The problem is to find |f'(0)|. There is no formula in any reasonable case.
Of course this is not a theorem, because one cannot define what a "formula" is. Bothe quantities, the modulus of a ring, and |f'(0)| in the simplified problem are solutions of certain extremal problems. So one can write a "formula" involving sup over some class of functions.
Added on 9.19: I don't know why the question about a "formula" is important. There are reasonably good converging algorithms for finding moduli of rings, of course. The closest thing to a "formula" for a conformal map of a simply connected region that I know is described in the papers of Wiegmann and Zabrodin, for example, MR1785428. Perhaps this can be modified to make a formula for the modulus of a ring.