Here's an update:
It's been to an Audi shop for an entire week and they are none the wiser. They took off the non-standard wheels, put on standard wheels and say that "90%-95%" of the problem has gone away. They then put on the wheels from our Audi onto one of their cars and it was also vibrating at speed, so I guess it's got to be the wheels. Trouble is that their proposed solution was to replace all 4 corners at around £2,500 which I obviously won't be doing.
A couple of questions:
1. The non-standard wheels have slightly larger hub holes and have plastic bits on the inside. Could a faulty one of these cause the car to shake at speed?
2. The tyres and wheels have been looked at a lot and no defects have been spotted. The tyres themselves are budget ones, but have a lot of tread. Should I splash out and buy a set of tyres and see if that fixes it?
Basically I am at a loss as I don't have the cash to just go and replace all 4 wheels, especially at the quoted prices, but need to fix the car. The Audi has been from post to pillar and nobody seems to have a clue. I have seriously lost faith in mechanics. What would you guys do if you were me?