It doesn't seem so complex to me. The door William uses to get into the "bar" is connected to the prep room. Imagine if there are tons of prep rooms for each customer, and they're all next to each other. You just need a moving bar room that slowly shifts along the passage to connect to each door as the guest is ready to depart. Once everyone is on board, it then moves to an exit shaft and moves upwards until it gets to the ground level where it then connects seamlessly into the train as part of a train car. Then the train departs.
https://i.imgur.com/Wuji4Wv.gifv
Yup. The dressing rooms are on an elevator system that "loads" guests one by one.