Removing the integrated starter generator (ISG) in P2 hybrid electric vehicles (HEVs) reduces the overall cost of the powertrain, yet it comes with an additional control complexity of the engine-start process. An appropriate coordination control strategy between the internal combustion engine (ICE), drive motor, and clutch is necessary to reduce the time for the engine-start while constraining the vehicle jerk. A time-optimal coordination control strategy with a predesigned vehicle acceleration trajectory is proposed from results of nonlinear model predictive control (NMPC), which can reduce the time for the engine-start process and obtain a small vehicle jerk. A dynamic model for the engine-start process is first built. In this model, a cylinder-by-cylinder engine model (CCEM) is derived to describe the ICE dynamics. A Karnopp and Stribeck combined wet clutch model is proposed to handle effects of speed difference on the torque, and calculation problem of frequent switches when crossing zero speed point. Subsequently, the optimal control problem is formulated with selected states and solved with NMPC. Finally, a rule-based control strategy is extracted from results of NMPC. Simulation results show that the engine-start process can finish within 260 ms, 13.3% less than the previous study, with a predesigned vehicle acceleration to ensure the vehicle jerk within 10 m/s3 and the peak to peak vehicle acceleration within 0.2 m/s2 in the case study under investigation.