From the DVD commentary, episode 3:
Douglas Mackinnon: Little bit of magic. If you're careful and you look at the beginning of that shot you'll discover that Aziraphale isn't in the car. And then suddenly he is.
Neil Gaiman: Did we magically CGI him or was he just sort of bending down?
Douglas Mackinnon: We CGIed him out. And the other thing that we should know about this car is that the morning that we shot this we - well it was David Tennant - broke the car. He broke the door on Aziraphale's side, so you couldn't get in or out of it anymore. It was stuck in, and there was actually a broken window.
Neil Gaiman: Which we had to CGI out.
Douglas Mackinnon: Yeah. So, at the end of the scene - although Aziraphale appears to get out of the car - he can't.
Neil Gaiman: Because he couldn't open or close the door.
Douglas Mackinnon: It was a directing cleverness to get him out.
UPDATE OCTOBER 2020 - the true story
Jeremy Marshall-Roberts, the owner of Mary the Bentley, at the ineffable con 2020: Somebody asked the question about the damage to Mary. Unfortunatelly, we overran, and Rob my stunt driver had already booked a holiday and off he went and so when he returned in January, on the 10th of January, I had this new driver who really had no clue how to drive old cars, so I showed him around, I showed him to go around corners. He came around the corner, the door was not closed properly for some reason, the door flew open as he went around. And instead of slamming on the brakes which is extremely efficient and would stop him straight away he kept on going, hit another car and really smashed the door quite badly. ... It did take the car off roads for 10 months. The door was completely remade because of this accident... it cost the total of £24 000 to rebuild the car as it is today.