How does it get to the H&C line?
The buggy goes to one of the higher numbered platforms at Paddington (12 perhaps) and then up a very steep, vehicle-only, gated slope, to high, level ground, where the taxi rank is. Then on to the (relatively new) entrance to the H & C line, where there is a lift down to platforms 15 and 16.
The buggy tends to appear up to 5 minutes after the
HST▸ has arrived, probably to give the alighting passengers time to disperse. After a few anxious moments of waiting, it will arrive. Oh, and driver is happy to accept a tip!