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Nonlinear time-optimal model predictive control (TOMPC) enables highly performant control of constrained systems, but its computational cost prohibits real-time use on fast dynamical systems. Neural network imitation of the TOMPC control law can reduce online computational costs, yet existing work mostly targets quadratic cost formulations in simulation and addresses the challenges of neural controllers in isolation. This work imitates a long-horizon nonconvex TOMPC with hard terminal constraints, whose high computational cost limits training-data generation. We present (i) \emph{self-informed sampling (SIFS)}, a data-efficient expert-sampling strategy that uses a learned time-to-go to query the expert only where the policy deviates from time-optimal behavior, (ii) a complete training and control architecture that jointly addresses dataset correlation, suboptimal expert labels, covariate shift, constraint satisfaction and model mismatch, and (iii) real-world validation on a Furuta pendulum. SIFS matches standard dataset aggregation performance in roughly one-third of the training time and with less than half the training data. The resulting controller matches the region of attraction of the expert TOMPC while raising the control frequency from 1.4~Hz to 1~kHz. On hardware, the controller swings up and stabilizes the pendulum along trajectories closely matching the expert TOMPC solution.