This page details some of the suppliers where I find bits for electronics and making in the UK. It is by no means comprehensive and I will happily add suppliers that you think rock if I have missed them off. I will also link to a similar list for another country if someone wants to make one.
It is great to buy UK if you can, but the fact is sometimes, e.g. when you need a large number of LEDs, you might just have to buy from China or elsewhere on the basis of affordability. Always check if the stuff you are buying is CE marked if it is going into something public and try and use reputable suppliers if you can. I have been pretty shocked at the shoddiness of quality and safety design in some of the things I have received from overseas before, but if you do order from overseas you will need to check the import duty/VAT that you may get stung for if the value is over the threshold. The current VAT thresholds are gifts worth more than £34 and other items worth more than £15, and duty threshold is £135 on gifts or other goods, but the rates change depending on the type of goods so its best to check the Gov website.
Regarding the cheap third party Arduino boards that you can find online, some fall into the realm of cheap knock offs and some are actually trying to do something that Arduino doesn't already do. Arduino have done a good thing and its good to support them, there is something to be said for buying at least one original board if you can afford it! Some of the cheap knock offs do have issues, I have seen some that require extra (undocumented!) software installs, some that had counterfeit serial chips in and got bricked and some that come in a kit with a load of jumper wires that were badly made and didn't even work. This stuff is hard enough anyway without being let down by faulty hardware, do your research and use a reputable supplier where possible.
http://www.bitsbox.co.uk - very cheap for basic bits, where I go to for solder, wire, resistors etc. To be honest I have had some other components that were a little bit too cheap, e.g. 3.5mm jack sockets that didn't last long, so maybe stay away from cheap high stress components!
These guys stock more physical computing bits like Arduino, sensors, Adafruit and Sparkfun stuff.Adafruit/Sparkfun has exceptional resources, tutorials and great hardware, but is US so one of these sites should stock the bit you want! :
https://www.coolcomponents.co.uk - likewise
https://shop.pimoroni.com - not really used these guys, they are quite new and setting themselves up as a maker paradise with a nice looking website, might do well to compare the price with others.
https://www.kitronik.co.uk - Aimed at education, big eTextiles section with interesting stuff in, lots of conductive materials and fun soldering kits, good bulk buys and sell making materials like perspex, foam, laser ply etc. Also lots of lesson plan ideas and information. Seem to be the place to go for micro:bit stuff.
https://makersify.com/ - Only just come across this, at first glance looks like a great selection of stuff for makers/ hobby electronics for reasonable price, will be trying them out!
These are more industry guys, great for buying bulk and Rapid has some great value tools etc always worth looking here for cheaper prices, but the websites can be confusing and the descriptions of stuff and how to use it can be lacking:
http://www.rapidonline.com - great for tools and general components
http://uk.farnell.com/ - boy, do they have a lot of stuff! Can be hard to find what you want, seems to be aimed at people with an electronics degree!
http://www.mouser.co.uk/ - I've not really used Mouser, they are big over the pond and have a UK branch too, YMMV.