The speakers in my 2012 Mini Cooper S go silent at somewhat random times--usually after I have driven 15-20 minutes. Sometimes it just goes silent with all the visual radio functions still working, sometimes it fades out, back in for a minute and then goes silent. Only once have I had it come back on while I was driving. Otherwise, if I stop and park for awhile (at least 15 minutes, I think), it will work again for 15 minutes, or so. I did have a couple days last week where it kept working for my entire drive of 20-30 minutes.
First thing I would check is the various plugs in the system that are easily accessible. Being it's a 2nd gen car I'm just not familiar with the plug locations.
The R56 has somewhat of an odd radio. As far as where to check the plugs on the radio I haven't a clue. Hopefully someone who has had their radio out can give you some guidance. If all your speakers are going off I'd suspect the radio & not the speakers. If it was a bad speaker I'd figure only one would be dropping in & out, not all of them.
Do you have the HK stereo or the basic radio? It could be the HK amp is going into some type of thermal shutdown until it cools off to a safe operating temperature. Not 100% sure about anything R56 but that sounds logical and I highly dought it's your speakers.
It's got to be a loose connection or over heating amp. I would think even the base R56 unit has a separate amp. So you've got connections at the back of the head unit, the amp, and the speakers. I know in my car I store stuff where the amp would be. If the amp was there it would got hot. An additional thought I hope isn't it. A short happening from a wire rubbing metal.