I don't know what the prices are in the U.K. versus the U.S., but a number of cameras come to mind and you've already mentioned some of them. The Panasonic LX7 punches above its weight, has an extremely fast zoom lens and is a joy to use. I just bought one myself as a daily go-to camera because the price dropped below $300 here in the States.
You certainly can't go wrong with the Fuji X10 if you can find one in your price range. The camera's sensor is a bit bigger than most of the competition (but not as big as that in the more-expensive Sony RX100) and the built-in viewfinder is great to have when you're shooting in bright sunlight. Plus it's a beautiful, cool camera. The XF1 is a decent little camera as well but its lens isn't nearly as fast as the X10's.
The Olympus XZ1. This camera is a bit older then the others but still does good work up to ISO 800. Its fast lens helps you keep the ISO settings low.
I mention this next camera last because pocketability seems to be imporant to you. But the Olympus E-PM1, or Pen Mini, is going for great prices these days. This micro four-thirds camera has the original 12mp sensor (as opposed to the more modern 16mp sensors). But the sensor is still so much bigger than what you'll find in all of the other cameras I've mentioned so far that it doesn't matter.
Olympus's intelligent auto mode works extremely well and the E-PM1's auto white balance is about the best I've seen, which means it can be a great point-and-shoot camera. There aren't a lot of exterior dial controls but the menu is very easy to manipulate.
The size of the E-PM1 body is not much bigger than the LX7 and XZ1, and certainly no bigger than the X10. But you have to mount an interchangable lens on the body, which can hamper pocketability. The camera comes with a collapsable 14-42mm f/3.5-5.6 lens (28-84mm equivalent) which really isn't bad at all. It's a manual zoom (the X10 is as well). You can always look around for a used Panasonic Lumix 14mm f/2.5 pancake. When you mount that on the E-PM1 the enitre package is about the size of the other cameras. I do this often with my E-PM1 and I can slip it into a jacket pocket.
But you have to keep in mind that a micro four-thirds camera is an entire class higher than the other compacts I've mentioned, as good as they are, with corresponding improvements in image quality. Even the kit zoom for the E-PM1, despite being technically slower than the lenses on the other cameras, really isn't an issue because you're still catching a good amount of light with the larger sensor.
Good luck. I hope this helps.