I had a similar problem with an Opel Manta years ago, although that was rear wheel drive and as such the problem was diagnosed as the Uj joint on the drive shaft. Problem would only occur between 60-90mph over 100mph and the problem went. This was of course on a track
Not sure what drive the TT has? I would have thought rwd or if it's a quatro? Just seems the same problem as I had, but then tyre's seem the most likely place to start?