We study superconductivity of twisted bilayer graphene with local and nonlocal attractive interactions. We obtain the superfluid weight and Berezinskii-Kosterlitz-Thouless (BKT) transition temperature for microscopic tight-binding and low-energy continuum models. We predict qualitative differences between local and nonlocal interaction schemes which could be distinguished experimentally. In the flat-band limit where the pair potential exceeds the band width we show that the superfluid weight and BKT temperature are determined by multiband processes and quantum geometry of the band.