There are two ways to approach this problem:
1. Browser detection
2. Responsive design
Browser detection usually works by analyzing the browsers user agent (either server-side in PHP or via Javascript) and then redirecting to a mobile friendly version of the site (usually at a subdomain, like your instance m.dvw.org.uk). If you were using PHP to do the detection you could internally route and output the design without the redirect keeping the URL the same for both sites (keeping the URL the same is a plus).
The downside to browser detection is that it is not always accurate, especially if the user disables sending the user agent string to the server.
Responsive design involves crafting your HTML and CSS to change for different devices (usually using screen sizes). This has the plus of keeping the URL the same, and not relying on the user agent string - only the capabilities of the browser. These type of website designs can be more complicated to design, as you have to make the same markup adapt to each device.
As of right now there isn't really a good way to handle responsive images. Images that work good on a desktop, might be to big for a mobile site and using a smaller image could save precious network bandwidth for mobile users - but that smaller image wouldn't work good on the desktop. Having a separate site via browser detection can handle this fine, by just outputting a different image. There are some tricks for responsive design, and it looks like there is a specification in the works, but don't expect broad browser support for a while.
I would really look into the responsive design strategy, of which you can find numerous articles online. Responsive design has the added benefit of having the website work well on the desktop even if the user re-sizes the browser.