Johnson's BMW 5 Series has an X-reg numberplate (September 2000 - February 2001), however it is the pre-facelift model, the facelifted version was released in late 2000. The episode was released in 2003, so contrary to what Sophie says it is not a brand new Beemer. The registration seen in the episode belongs to another car, it's a common practice for films and TV programs to use fake or other cars' registrations for prominently featured cars.
When Jeremy is driving Johnson's BMW, the hand of someone sitting in the back seat is visible when the camera points at Mark (Jeremy's POV). Only Mark and Jeremy are supposed to be in the car.
When Mark leaves the hotel, he walks down the side of the road trudging through mud. Yet there's a pavement clearly visible on the other side of the road, there's no need for him to be walking through the mud.