Instructions for Ubuntu:

1. Download the latest .tar.bz2 WebRunner package for Linux and extract the webrunner folder somewhere permanent (I use ~/local for this kind of thing).
2. In Terminal, run this command (with the path edited as appropriate):

   /path/to/webrunner/webrunner -uri "http://www.bbc.co.uk/radio/aod/mainframe.shtml"

3. If that worked ok, open Ubuntu's System > Preferences > Main Menu editor and add a new item to the Sound & Video section, with the name BBC Radio Player and the command as above.
4. Enjoy the BBC Radio Player running as a separate application, independent of Firefox.

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Update: here's a .webapp file that you can run from the command line:

webrunner -webapp bbcradioplayer.webapp

(or by double-clicking on some platforms) that will open up the BBC Radio Player in WebRunner. You can then use the menu in the bottom right-hand corner to save an icon on your desktop for launching the application.

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Update: On OS X it's even easier - just download the WebRunner .dmg, install and run it once to register it as a .webapp handler, then open the .webapp file above. Saving an application bundle to the desktop doesn't seem to work yet though...