Quantum key distribution (QKD) enables secure communication even in the presence of advanced quantum computers. However, scaling up discrete-variable QKD to high key rates remains a challenge due to the lossy nature of quantum communication channels and the use of weak coherent states. Photonic integration and massive parallelization are crucial steps toward the goal of high-throughput secret-key distribution. We present a fully integrated photonic chip on silicon nitride featuring a four-channel wavelength-division demultiplexed QKD receiver circuit including state-of-the-art waveguide-integrated superconducting nanowire single-photon detectors (SNSPDs). With a proof-of-principle setup operated at a clock rate of 3.35 GHz, we achieve a total secret-key rate of up to 12.17 Mbit/s at 10 dB channel attenuation with low detector-induced error rates. The QKD receiver architecture is massively scalable and constitutes a foundation for high-rate many-channel QKD transmission.