Mine was pretty quick today given the heat. Was regassed last year, had a new sensor within the last two (as it had gone porous), and I know the compressor was replaced under warranty before my ownership. Was bliss driving home after the crunking work aircon 'tripped out' - which it always seems to when it is hotter or colder than usual!
Long answer to a short question - yes, mine was cooling within about 20 seconds. Was making plenty of noise as it adjusted the various flaps in the dash, but guess that was all the vents and such being set. It's a noisy system, that's for sure...