Bilayer graphene at certain small internal twist angles develops large scale moiré patterns with flat energy bands hosting correlated insulating states and superconductivity. The large system size and intricate band structure have however hampered investigations into the properties of the superconducting state. By using full-scale atomistic modeling with local electronic interactions, mimicking closely those of the high-temperature cuprate superconductors, and solving fully self-consistently for the superconducting order, we find a highly inhomogeneous superconducting state with nematic ordering on both the atomic and moiré lattice length scales. More specifically, we obtain locally anisotropic real-valued d-wave pairing with a nematic vector winding throughout the moiré pattern and a three-fold degenerate ground state. Despite the d-wave nature, the superconducting state has a full energy gap, which we show is tied to a $\pi$-phase interlayer coupling. We further show that the superconducting nematicity is easily detected through signatures in the local density of states. These results show both that atomistic modeling is essential for superconductivity in twisted bilayer graphene and that the superconducting state is distinctly different from that of the cuprate superconductors. in twisted bilayer graphene and that the superconducting state is distinctly different from that of the high-temperature cuprate superconductors, even if their electronic interactions may be the same.