I also think there is no "explicit formula" for the conformal modulus, even in simple cases. However, as mentioned in Igor Rivin's answer, there are methods for approximating the conformal map and the conformal modulus. These methods are generelizations to doubly-connected domains of the well-known Bergman Kernel Method for simply connected domains.

For an introduction to these methods, I suggest you take a look at this document, sections 1.3 and 2.5.

I also think there is no "explicit formula" for the conformal modulus, even in simple cases. However, as mentioned in Igor Rivin's answer, there are methods for approximating the conformal map and the conformal modulus. These methods are generelizations to doubly-connected domains of the well-known Bergman Kernel Method for simply connected domains.

For an introduction to these methods, I suggest you take a look at sections 1.3 and 2.5 of Lectures on Numerical Conformal Mapping by Nicolas Papamichael.