Mine's only the 520d (2012 with Sport Auto) but the lag is a nightmare from a standstill and on the move.
BMW won't accept it's a problem though. My dealer says it drives like that on all of them and is apparently down to the torque converter disengaging when it's stationary - the delay is the time taken to re-engage.
The throttle delay when the car's still moving is because the gearbox wants to keep as high a gear as possible for as long as possible to keep the MPG high - comfort or sport mode.
All sounds plausible but as far as I'm concerned the throttle delay ruins an otherwise fantastic car. It's unlikely I'll have another one.