The point at which the turbo "spools up" (begins working) is a function of many things, su...
The point at which the turbo "spools up" (begins working) is a function of many things, such as: turbo size, plumbing, size of the motor, exhaust manifold, etc.
A turbo is driven off of exhaust gases. The 300D's don't provide much boost up to 2,500rpm mainly because there is not enough exhaust to create full boost. At around 2,500rpm the engine is pushing enough exhaust to fully spool the turbo. This is quite common and was almost universal untill recently with twin turbos and modern computer controlled fuel injection. You may have heard it refered to as "Turbo Lag". This is especialy prevalant in cars with single turbos. The lag is just what I mentioned before - the "down time" before the turbo kicks in or creates full boost. Even today, many modern turbo cars still have some degree of turbo lag. For example, the Subaru WRX doesn't "kick in" or create full bost untill 4,000rpm! This is because the car runs a 1.0 bar (14.7 psi) of boost pressure and the engine is only a 2.0L 4 clyinder.
I am guessing that the audi you drove was the S4, Allroad or A6 2.7? All of these cars have the same engine, a 2.7L V6 Twin Turbo. The engine has one turbo for each bank of cylinders, that along with the modern computer fuel control makes for an almost seemless turbo powerband.