An on-demand source of bright entangled photon pairs is desirable for quantum key distribution (QKD) and quantum repeaters. The leading candidate to generate such pairs is based on spontaneous parametric down-conversion (SPDC) in non-linear crystals. However, its pair extraction efficiency is limited to 0.1% when operating at near-unity fidelity due to multiphoton emission at high brightness. Quantum dots in photonic nanostructures can in principle overcome this limit, but the devices with high entanglement fidelity (99%) have low pair extraction efficiency (0.01%). Here, we show a measured peak entanglement fidelity of 97.5% ± 0.8% and pair extraction efficiency of 0.65% from an InAsP quantum dot in an InP photonic nanowire waveguide. We show that the generated oscillating two-photon Bell state can establish a secure key for peer-to-peer QKD. Using our time-resolved QKD scheme alleviates the need to remove the quantum dot energy splitting of the intermediate exciton states in the biexciton-exciton cascade.