The theme for the weekly competition was to have space aliens race against each other. And it was all meant to be a spoof. These figures were not meant to be the real characters.
DARTH VADER - Piss-poor costume was a give-away. We're not meant to see his neck but if we did, it should have been horribly scarred. Besides, the real Darth Vader (aka Anakin Skywalker) died a long time ago in a galaxy far, far away.
MING THE MERCILESS - Only a few years ago live-action TV finally got its own version of 'Flash Gordon', but I refuse to accept it as part of Earth Prime-Time. Or at the very least, it'll be part of the main Toobworld only until a version more in keeping with the look of the original movie serials comes along. This Ming looked the part, moreso than the surfer boy in velcro pajamas from the short-lived TV series, but it was obviously just a guy running around in a cape.
THE KLINGON - It was tempting to make the claim that this was a real Klingon, but it would go against the established canon's timeline (not that it's ever stopped me before!) to have a Klingon make contact with Earth so early. I'm going to side with the opinion that this was a Trekkie in costume.
THE DALEK - This was a pretty convincing model, but that's all it was. Had it been the real thing, then all of those people in the TV studio would have been dead.
THE CYBERMAN - This was the Doctor's target. And here's how I think it played out:
In that alternate Sixth Doctor's home dimension, John Lumic was just bringing his version of Cybermen to "life". There had to be a few test models before the Cybermen were ready to go into mass production.
The Cyberman on 'Top Gear' was one of those top models, not yet fully programmed to bring humanity to its knees. Its programmable engrams were still quite pliable, so when Jeremy Clarkson told it to race around the trick (thinking it to be just an actor in a suit), it did so.
How did it get to Earth Prime-Time? It may have been tagged along with the alternate Doctor, hitching a ride outside the TARDIS in much the same way as Jack Harkness did with the Doctor of the main Toobworld. But it wasn't traveling a great distance through Time, only through a dimensional vortex. However, just before the TARDIS arrived on the main Toobworld, the Cyberman fell off and landed a few years into the Past.
Giving chase, the Doctor got himself inserted into that mock competition in order to curtail the threat posed by the Cyberman without alarming the innocent people nearby.
The Doctor deliberately came in fourth, and more than likely used his sonic screwdriver to skew the results for the other contestants so that the Cyberman came out on top.
Also happening off-screen - the Doctor must have adjusted the "trophy" given to the Cyberman so that even though it didn't affect ordinary humans, the Cyberman's circuitry was deactivated as soon as it came into contact with the trophy.
After deleting the Cyberman's programming so that it was no more than a metal scarecrow, the alternate Sixth Doctor returned to 2006 at just the right moment to be "swapped" back to his rightful place in that alternate dimension.