You need to start by narrowing this down to which belt is generating the noise, remove the serpentine belt and start it up. If the noise is still present then you have a timing belt issue, if not then something in the serpentine belt circuit is shot. If this is the case then while you still have the serp off give all of the pullies a few spins apiece to see if it can be narrowed down this way. If not then you need to put the belt back on, fire it up, and then carefully use a mechanic's stethoscope to try and locate the noise. If the noise turns out to be in the timing belt circuit then I would recommend the shotgun approach, go in there and replace the idler and tensioner pullies, if the tensioner shows signs of leakage then replace that sucker too. I guess it is possible that the water pump could be making the noise but it would be unsual and not consistent with how your noise presents itself, typical bad water pump bearing noises do not come and go like yours does. Most of the bearing noises that I have had that only duplicate on a cold engine for a minute or two when first started have been timing belt tensioner or idler pullies.

Mark