Reliable nowcasting of extreme precipitation remains difficult because convective systems are strongly nonlinear, multiscale, and nonstationary in 3D. Radar is the backbone of nowcasting, yet existing methods struggle to predict extremes: physics-based extrapolation cannot capture growth and decay, deterministic learning tends to oversmooth and underestimate peaks, and purely generative models often lack physical consistency. Hybrid schemes help but are mostly limited to 2D composite reflectivity, collapsing the atmosphere into one layer and discarding vertical structure critical for height-dependent dynamics. We introduce Nowcast3D, a gray-box, fully 3D framework that works directly on volumetric radar reflectivity. The end-to-end model couples physically constrained neural operators (advection, local diffusion, and microphysics) with a conditional diffusion model to generate ensemble forecasts with quantified uncertainty. Trained on provincial-scale 3D volumes over a $10.24^\circ \times 10.24^\circ$ region and fine-tuned on a $2.56^\circ \times 2.56^\circ$ city region ($0.01^\circ \approx 1$ km), Nowcast3D provides near-real-time forecasts up to 3 h and outperforms competitive baselines in cross-region and temporal out-of-sample tests. It can also infer wind fields without labeled supervision, supporting physically plausible transport. In a nationwide blind evaluation by 160 meteorologists, Nowcast3D ranked first and was preferred in 57% of post-hoc assessments, surpassing the leading baseline (27%). These results highlight its reliability and operational value for extreme precipitation nowcasting.