Install mplayer plugin for firefox, if mplayer has problem with bbc (it hasn't for me), just install the original
Realplayer from the websites' .bin and copy/paste url from Mplayerplugin window (right clic on the windows, then select "copy url") to Realplay.
Since Realplay .bin is static it works on x86-64 too.
Alternatively, bbc has a flash version of it's client "bbcplayer" (or whatever it's called...), search their website to launch the player ui and launch the radio from there.