Quadrotors are widely employed across various domains, yet the conventional type faces limitations due to underactuation, where attitude control is closely tied to positional adjustments. In contrast, quadrotors equipped with tiltable rotors offer overactuation, empowering them to track both position and attitude trajectories. However, the nonlinear dynamics of the drone body and the sluggish response of tilting servos pose challenges for conventional cascade controllers. In this study, we propose a control methodology for tilting-rotor quadrotors based on nonlinear model predictive control (NMPC). Unlike conventional approaches, our method preserves the full dynamics without simplification and utilizes actuator commands directly as control inputs. Notably, we incorporate a first-order servo model within the NMPC framework. Through simulation, we observe that integrating the servo dynamics not only enhances control performance but also accelerates convergence. To assess the efficacy of our approach, we fabricate a tiltable-quadrotor and deploy the algorithm onboard at a frequency of 100Hz. Extensive real-world experiments demonstrate rapid, robust, and smooth pose tracking performance.