Supercharger is always on, its linked to the engine by a belt ussually so as long as your crankshaft is turning, your supercharger is sucking down the air

The turbo is exhaust driven so it will spool up later on in the rev range, i dont know the figures off hand im afraid. but if theyve designed the twincharge system right the supercharger gives you an immediate boost in air intake which compensates for any turbo lag.
I used to run a cooper S with the supercharger (not like the pretenders later with the turbo charger!)
Thing sounded glorious especially after i added a modifed K&n filter and airbox...
Drank petrol like it was going out of fashion though
I've got the new 1.4 ACT engine, its only turbocharged to give 140bhp... but i honestly dont notice any turbo lag... though comparing it to the mini was like apples and satsumas...