This paper offers a new algorithm to efficiently optimize scheduling decisions for dial-a-ride problems (DARPs), including problem variants considering electric and autonomous vehicles (e-ADARPs). The scheduling heuristic, based on linear programming theory, aims at finding minimal user ride time schedules in worst-case quadratic time. The algorithm can either return feasible routes or it can return incorrect infeasibility declarations, on which feasibility can be recovered through a specifically-designed heuristic. The algorithm is furthermore supplemented by a battery management algorithm that can be used to determine charging decisions for electric and autonomous vehicle fleets. Timing solutions from the proposed scheduling algorithm are obtained on millions of routes extracted from DARP and e-ADARP benchmark instances. They are compared to those obtained from a linear program, as well as to popular scheduling procedures from the DARP literature. The proposed procedure mostly yields optimal solutions, with nearly-optimal solutions occurring in only 27 out of 21.5 million cases on DARP instances. Additionally, it demonstrates an average speed improvement of around 60% compared to a linear program and performs comparably to benchmark scheduling approaches from the DARP literature, while outperforming them in solution quality.