Okay, I'd suggest one of the ones that I mentioned in the group test above.
The Sony RX100 in the review and mentioned by daveillot is the absolute best advanced compact you can buy, in a class of its own compared to the rest, but it's over your budget at £400 new/£350 used.
The Olympus XZ-2 again mentioned by davelliot is also a good pick and can be had for just slightly over £300.
A better choice, IMO, is the Panasonic LX7 - this has a slightly shorter maximum zoom than the XZ-2, but a wider wide-angle, meaning you can fit slightly more into your photos. It also has a slightly brighter lens which means it'll be a little better in low light. Also just slightly over £300.
A little cheaper is the Nikon P330 - this has the same sized sensor as the Olympus and Panasonic and will give you the same quality photos in good light, but as you zoom in the lens closes and lets less light in, meaning the zoom isn't as good in low light. But with no zoom it's just as bright, and this is only important in low light. On the plus side it's smaller and lighter. £220 from Amazon.
Alternatively there's the Nikon P7700, which has the same sensor as the P330 but has a better lens that maintains its brightness as you zoom in. The price you pay is that it's chunkier and heaver. £300 from Amazon.
Another option is the Fujifilm XF1. This is a small camera like the P330 and the lens has the same limitations - it's not as bright at full zoom in low light, but great in all other conditions. On the plus side, its sensor is bigger than all except the RX100. And it has some handy film simulation modes that tweak the colour and contrast of your photos, making them look great straight out of the camera. ~£220 from Amazon.
All of these cameras have manual controls, but they can just be shot in auto mode, so they're all user friendly.