To address the overfitting and unreliable uncertainty estimation of purely data-driven approaches, this paper proposes a hybrid physics-data framework (Phys + GPR) for battery prognostics. First, a three-segment empirical model, derived from the early, accelerated, and linear degradation stages of lithium-ion batteries, is employed to extract a physics-based capacity prior. The residuals between measured capacity and the prior are then modelled by a two-stage heteroscedastic Gaussian process regression (GPR), Stage 1 estimates the residual mean, Stage 2 estimates the input-dependent variance. A TreeBagger random-forest regressor further refines the mean prediction, and β-calibration is applied on the training set to scale the predictive intervals, ensuring a reliable 90% coverage throughout the battery lifetime. Leave-one-battery-out (LOBO) cross-validation on NASA cells B0005, B0006, B0007 and B0018 shows that Phys + GPR achieves R2>0.93 for all cells, with a 90% prediction-interval coverage probability (PICP) of 70%~92% and a mean prediction-interval width (MPIW) of 0.085~0.10 Ah—significantly outperforming pure GPR, single-exponential + GPR and SVR baselines. The results demonstrate superior cross-battery generalisation, interpretable physics priors and robust uncertainty quantification, providing high-confidence support for battery health management and online RUL prediction.