There seems to be some misinformation floating around about what transmissions are used in which model of the TT, so I'll take a moment to set it straight.
The 180FWD uses the 02J transmission. The 02J is an older and weaker design than the 02M. The 02J from the Audi TT FWD and other drivetrain components (axles, clutch, flywheel) can be swapped directly into a 1.8T MkIV Volkswagen and vice versa. I sold my old 02J to a local guy with a GTI who is now running it.
All Quattro cars use the newer 02M transmission. The 02M is a newer design and is generally bulletproof. For a high HP drag car, an 02M is rather desirable as the 02J has a habit of breaking from abuse.
The 225 uses the same type of transmission as the 180 Quattro, just with slightly different ratios and a sixth gear added. IT IS NOT a better/stronger transmission than the 180 Quattro's.
It is possible to swap a 6-speed 02M from a Jetta GLI or 337/20th AE GTI into a FWD TT in place of the 02J, however it's not cheap or simple. Apart from the transmission itself, which costs $1000-$1500 used, one must obtain axles, a clutch and miscellaneous pieces which total up to around an extra $1000 (going with used parts, mind you) on top of the cost of the transmission itself.
Here's a chart denoting the ratios for each: