Hi, I'm a new member here and don't even own a TT yet (I'm looking) but I did see a Wheeler Dealer episode that addressed the issue of failed 1st and 2nd in a 6 speed Quattro TT.
Go online and look for Season 11 Episode 10. Ed China (the shows mechanic) figures out how to fix the problem cheap. He did have to drop the transmission and open it up, but not tear it down. The problem is a failed rivet that holds a selector fork.
If you're lucky like the car in the show, the part has only fallen to the bottom of the gearbox and not damaged anything. Ed was able to fix the problem with a new bolt.
While you will have to still find a good shop near you, show them this episode and maybe save some big bucks.
In the mean time don't drive the car! Good luck!