I'm not sure that this is really a "prod" on the bars - more of a controlled push - the movement is quite minimal in fact. As you say, its the speed that this is achieved which dicates how quickly you turn.
I struggled with this for a while, until I realised that I was in fact tensing up on the bike before I initiated this (probably as a result of braking) and resisting this movement by tightening my grip on both bars and actually controlling (or preventing) the movement of the inside bar, by gripping the outside bar too tightly.
At a recent CSS day I cured this by actually moving my hand away from the outside bar completely (just a cm or so, whilst still covering the bar) and only applying pressure to the inside bar. Didn't help much with the pickup though!!!!
HIH
Tim