Laboratory laser experiments offer a novel approach to studying magnetized collisionless shocks, and a common method in recent experiments is to drive shocks using a laser-ablated piston plasma. However, current experimental capabilities are still limited to spatiotemporal scales on the order of shock formation, making it challenging to distinguish piston and shock dynamics. We present quasi-1D particle-in-cell simulations of piston-driven, magnetized collisionless shock formation using the code PSC, which includes a model of laser-driven plasmas that can be well-matched to experimental conditions. The simulations cover a range of upstream and ablation parameters and yield several robust signatures of shock formation, which can provide a reference for experimental results.