Accurate prediction of creep deformation in zirconium (Zr) alloys is essential for structural components, such as nuclear reactor fuel claddings, particularly under irradiation environments. However, existing irradiation creep models often rely on phenomenological formulations or oversimplified mechanisms, limiting their predictive capability. In this work, we first establish a physics-informed, multi-mechanism creep framework for zirconium based on jogged screw dislocation dynamics, which are experimentally recognised as a key mechanism in the thermal creep of Zr. The model explicitly incorporates dislocation glide and climb mediated by jog dragging and jog bypass. Notably, both pipe diffusion and bulk diffusion of point defects are formulated within a thermodynamically consistent framework, which helps to quantify the role of pipe diffusion in dipole dragging and its associated contribution to thermal creep. A hybrid calibration strategy combining machine learning techniques and experimental creep data is employed to determine physically consistent parameters, allowing the reproduction of both transient and steady-state creep responses. Building upon this validated thermal creep framework, we further extend the model toward irradiation creep by coupling cluster dynamics with dislocation dynamics. The proposed extension accounts for interactions between dislocations and irradiation-generated defect clusters and prismatic loops, as well as defect-enhanced jog climb under point defects with high concentrations. This naturally takes climb-assisted glide into account as a deformation mechanism under irradiation. The unified framework provides a physically transparent pathway to bridge thermal and irradiation creep in zirconium alloys, offering a scalable platform for predicting deformation behaviours in extreme environments.