That really does sound like the secondary air pump mate cause it only lasts for a few minutes when its cold anyway, I would hazard a guess its got water in it at some point as mine did from the cracked plastic pipe that comes of the airbox, mine made this horrific screeching loud noise on my S3 & my S4 B6 had a noise to but hey that was a V8 so seemed normal, but the S3 one was far from normal, so changed out & noise went away.
Buy new cause then if goes again can get it swapped out within 2 years, expensive but so worthwhile if you really think its this, tell you what to do though, could try unplugging the loom from the pump & see if noise goes away, before starting up in morning, its fine to do for few minutes to test this theory.
Also mate I can now recall from memory on cold mornings the S3 was more smoky when I had the issues, now from this it would be my 1st port of call but upto you at end of day.