The glove box is a pretty common issue with these cars, it has a habit of having the latch break.
If you're lucky enough to get a replacement, here's how to go about removing your old one. You follow these instructions at your own risk.
1. First, disconnect the battery using proper procedure not to electrocute yourself or damage stuff. Unplugging the airbag switch might throw an airbag light, which I'm not sure whether you can turn off with a VAG scanner or not. Either way, I disconnected the battery as a precaution against throwing an airbag light and it worked.
2. Open your passenger door, pull off the panel that covers the passenger's side of the dash, it's the same as the driver's side that covers the fuse box. Proceed to remove the torx screws, I think I took off more than I needed to. Don't lose them.
3. Grab a ratchet extension and sneak your way to the two torx screws under the box. The little sill conveniently catches the screws as you remove them. As you can imagine, this sill is not on the new TT, Audi's engineers must have gotten pissed that they let something that was convenient to work on into production.
4. Open your box. If it's stuck, which is very likely considering you're taking it out for a replacement, you can reach in from the passenger's side and push in the little round piece that's causing it to get stuck. You may have to pull the black panel off to do this:
The piece should be in one of the holes above the metal speednut next to the glovebox door. Just get the box open somehow.
5. Remove the three torx screws at the top of the box, and remove it CAREFULLY there are still cables attached to it!
6. Looking at it, you'll see the box has the box has wires for the airbag key and light attached to it. They simply unclip; the airbag cable unclips upwards. The cable for the light unclips at the rear of the box.
7. Box should be free now:
8. Now you have to remove the airbag key module. Audi kinda melted some little plastic rivets to hold it in place or something. Either way, it's not too difficult to remove, but takes a bit of patience. Grab a sharp utility knife and CAREFULLY trim off the melted plastic. Keep your hands clear of the blade at all times.
Last one's a bit awkward to reach, but it's doable:
Once you've chopped off all the plastic pieces, you should be able to just push it right out (in the direction as if you were pushing it into the cabin of the car).
For your replacement glovebox, just remove its key module and glue yours in.
8. Make stupid jokes with your newly removed box.
"Excuse me while I get my registration out of the glovebox"
Installing new box is the reverse of above.