Puzzling noise when getting going

Hello fellow Copen owners. I've just bought my dream car, I'm chuffed to bits! It's an 08 so 1.3 engine.
I'll be posting stuff about all the remedial work that I intend to do on it, but let's start first with an intriguing issue I've found.
OK, so I start the car and everything is fine (other than the noise from the timing chain, but that's something for a separate thread). It doesn't matter whether I remain static for a while or I go straight away, as soon as the car starts to move forwards for the first time it would make a noise, more like a "quiet scream", for a second or so, kind of the sound that a battery-powered drill/screwdriver does when in operation.
Then, once it's done this, I can stop a traffic lights, traffic, etc. and it won't do it again until I stop the engine, let it rest for a short while and start it again... puzzling, uh?
I can't figure this out. Has anybody experienced anything similar? I'll keep you posted of progress (if any).
Cheers.

 

Wow, that was a very quick and extensive response. Many thanks guys/gals!
All good alternatives, although I'm more inclined towards Shane's suggestion as the sound is kinda electro/mechanic and only happens at the onset of every "trip". I'll investigate further and let you know.
Many thanks again.

 

Thanks all. I haven't heard this in any other car before. I managed to get it on camera. In between the two videos the engine was switched off and restarted. I guess it would let you more puzzled than before...
The noise can be heard as much inside than outside. Bear with me whilst trying to upload these videos...

 

Noise from outside:



Noise from inside (after restarting):

 

So, as mentioned the chain has been making noise for a while (not the one on this thread) and would need replacing. I've been quoted £250 just on parts for the chain kit. AND the engine has 130k miles on the clock. I've found a replacement engine with 45k miles for less than that and I've decided that the best avenue would be to replace the lot. Engine comes with all the ancillary components so chances are that both noises will disappear with the swap. Wish me luck! Tuesday my Copen goes under the knife...

P.S. Has any of you gone through an engine swap in your precious Copen? If so, any hints, tips, etc. Will be highly appreciated.

 

More on the sound. After replacing the engine the sound remained... , so I've followed Binz advice (safer than welz's suggestion ) and started pulling fuses and relays. It turns out the sound is caused by the ABS pump .
Now the fun begins to understand why it does it. In theory the pump would only energise when there is a loss of pressure after the ABS valve releases when ABS intervention is required. I haven't tried the breaks hard enough yet but will do tomorrow. I'll keep you all posted but in the meantime does anybody have any suggestion?