In reply to Martin Hore:
I think the Panasonic TZ80 is considerably thicker, front to back, than it's predecessors of which I've had several - TZ10, TZ27, TZ30. This may make it a bit unwieldy by comparison. Incidentally I had that many TZs (a) because I liked them, and (b) because I broke them easily! For the latter reason I went on to Canon and have a Canon Ixus 235 (this year's equivalent is Ixus 285 I believe) for compactness and a Canon Powershot SX270 for extra zoominess and manual controls.
The Ixus is what I mostly use when climbing - about the size of your current Nikon I think, but with 10x zoom, and I find it, for what it is, surprisingly good in low light, it's light in weight and it's very robust - seems to stand up to my carelessness, which probably did for the Panasonics!
Either of the above are half the price of the TZ80 (or "proper" cameras like the RX100 mentioned above). Or pick up a used Panasonic TZ30 or 27 or 35 for under £100.
Bear in mind I'm no photographer and also have no idea what might fit into the pouch you describe, but I do always carry a camera, climb long routes whenever I can and have been doing so for a very l-o-n-g time, so am used to the problems of keeping a camera handy and intact (though lamentably bad at the latter!).