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As far as I know all the Veyrons are made by order and a deposit is required. So technicaly the car is private even before it is made. And of course, they wont deliver a car that has been driven for a 50-100 km. Nonetheless, Top Gear menaged to make 5-6 videos with different Veyrons.

Manufacturers provide TG with factory demo vehicles, press mules and sometimes even prototypes. They want to promote their cars. Sometimes though cars are borrowed from friends of the show. Nick Mason's Ferrari Enzo for example or Rowan Atkinson's McClaren F1.

Top Gear did several shoot with the Bugatti, some of which were factory supported. It is clear that the ones with the factory support teams were a "Media Mule". The other shots may have been with a loaner from an owner, the credits are unclear.