Cam chain tensioner rattle in the video right? Mine does it as well. If you google it you'll find a lot of information on it. The tensioner itself is hydraulic so the oil pressure doesn't build up fast enough during the first start, meaning it doesn't tension the chain as fast. That produces the rattle, once the oil pressure builds up the rattle goes. It is a tricky subject. Some replace the tensioner and the problem remains. Issue could be in the oil pump or pipes also, so replacing a tensioner might not help. But it could. Mine does it only on cold startups on colder days. If it does however do it when revving and become more constant I'd get the tensioner replaced ASAP. It isn't the cheapest thing to replace and you should avoid cheap ones and knock offs like the plague.
Google "1.8T" and "cam chain tensioner rattle", and you'll see a lot come up. Even from audi-sport.net 8L section.
EDIT: I believe Audi themselves have released a statement about the rattle and how replacing the tensioner won't help and there is nothing to do about it and is common and normal (on cold startups and only for 1-3 seconds). Yet the problem is you never know how much noise Audi engineers find normal in this case and what not. So the statement isn't really anything solid but could give you a little peace of mind.
As you might notice in the video, it does not only do it for 1-3 seconds on start up but continues to go on. On my engine it keeps on knocking till the engine warms up and then stops.When i rev up the engine a little while cold the knocking goes faster following the revs. I noticed this summer it did not do it when the weather was warm so im thinking it might have something to do with the valves.