How long does the Nikon repair centre usually take to repair point-and-shoot cameras?

I think it's got a sticky shutter as when I turn it on, the lens zooms out but the screen is black and won't take photos. How long do these things usually take to fix as I'm going on holiday in less than 3 weeks…

Usually a month or more
depends on how many repairs they have

You should probably just buy a new camera - the repair for your current camera will more than likely cost more than what the camera is worth. Check out the Powershot SX150 and the Coolpix S6200 and L120.

A sticky shutter is usually caused by a buildup of lubrication oil on the shutter in a camera that's been sitting for a long time unused. But in most cases, this can easily be fixed by the owner without even opening the camera. See the following:
http://camerarepair.blogspot.com/2009/05/simple-fix-for-stuck-shutter.html