Adding to this conversation because I noticed Oldkool is in Madison, Wisconsin, USA. This is all music biz minutia but if you're interested, here it is. Are you watching Top Gear on BBC America or a foreign feed?
First, the use and distribution of any copyrighted music in a TV show (or any other medium) is subject to the permission of the copyright owner. This is theoretically true throughout the western world. However, BBC Television (UK) has the afore-mentioned blanket agreement with a British performing rights society called PRS. It grants BBC TV that permission for any music represented by PRS (which is a large amount, but finite). Now, this is only for broadcast in the UK. Instead of paying individual licences to each copyright holder, BBC TV pays a sum fee for this right and saves millions.
No broadcasters have such rights here in the States. So, prior permissions, usually accompanied by heavily negotiated fees, are required for using music here. Good old American supply and demand dictates that the more popular the song, the more expensive it will be to use. So, stating the obvious, that's why you hear "knockoffs" on the History Channel.
So here's where it all ties in: Though it's owned by BBC (UK), BBC America is an American network (just like History Channel) and BBC TV's UK blanket agreement doesn't cover broadcasting PRS music here. Chances are, if you're watching Top Gear on BBC America (or DVD), you're hearing some of the music originally broadcast on the BBC (UK), but you're also actually hearing a lot of knockoff music that's replaced the original music in editing. For practical or creative purposes, the producers will have wanted to retain as much of the original music they can afford - and they can probably afford a lot on Top Gear's budget. For instance, the theme song "Jessica" has become an iconic part of the show in the UK, so they were willing to pay the hefty licensing fees to preserve it here in the States. It's probably not so with some of The Stig's driving sequences.