I am looking for some advice on what wide-angle lens to get for the D3000. At the minute I have seen this 10mm lens but I am unsure whether it will work well with the camera in question. Ideally I would like to be getting a lens within the £200 mark and below the 18mm that comes with the stock lens.
No, that lens won't work with a D3000, because it's for the new "Nikon 1" system, which uses a smaller sensor and a different (smaller) lens mount. (And, because of that system's smaller size, note that the resulting angle of view is almost exactly what you see through your kit lens at 18mm.)
Your best bet for a budget ultra-wide lens for your camera may be the Samyang 14mm f/2.8. This will run you around £300, and is manual focus only, but is fairly well-reviewed for its price. Another option might be the Sigma 10-20mm f/4-5.6 zoom. It's not as fast, but will autofocus (neither of which are a big deal in an ultrawide in my opinion). Like the Samyang, reviews note complex distortion and other artifacts, but it still manages to come in at under £350.
The fact is that good wide angle lenses are intrinsically expensive. So, another option is to be happy with the 18mm of your kit lens until you can save up a bit more. I was in the same situation and got a cheap wide-angle zoom, which in retrospect could be considered wasted money better saved towards the nice ultra-wide I eventually got (a Pentax 15mm — no help to you!). On the other hand, getting something let me get images right away, which is arguably infinitely better than not having anything.
The Tokina 11-16mm is another good, sharp lens. A used copy might be close to your budget