Based on the mathematical theory of isotopic tilings on hyperbolic surfaces and mapping class groups, we present the, to the best of our knowledge, first algorithms for the enumeration of isotopy classes of tilings by compact disks with a given symmetry group on hyperbolic surfaces, which is moreover combinatorial in nature. This enumeration is relevant for crystallography and materials science. Using the theory of automatic groups, we give some results on the computational tractability of the presented algorithm. We also extend data structures for combinatorial classes of tilings to isotopy classes and give an implementation of the proposed algorithm for certain classes of tilings and illustrate the enumeration with examples.