Over the past decade, there has been a remarkable surge in utilizing quadrotors for various purposes due to their simple structure and aggressive maneuverability, such as search and rescue, delivery and autonomous drone racing, etc. One of the key challenges preventing quadrotors from being widely used in these scenarios is online waypoint-constrained time-optimal trajectory generation and control technique. This letter proposes an imitation learning-based online solution to efficiently navigate the quadrotor through multiple waypoints with time-optimal performance. The neural networks (WN&CNets) are trained to learn the control law from the dataset generated by the time-consuming CPC algorithm and then deployed to generate the optimal control commands online to guide the quadrotors. To address the challenge of limited training data and the hover maneuver at the final waypoint, we propose a transition phase strategy that utilizes MINCO trajectories to help the quadrotor 'jump over' the stop-and-go maneuver when switching waypoints. Our method is demonstrated in both simulation and real-world experiments, achieving a maximum speed of 5.6m/s while navigating through 7 waypoints in a confined space of 5.5m*5.5m*2.0m. The results show that with a slight loss in optimality, the WN&CNets significantly reduce the processing time and enable online optimal control for multiple-waypoint constrained flight tasks.