The BBC Media Player does nothing stand-alone, it's a "back end" to apps like the BBC iPlayer or BBC Radio Player, all of which were on my Nexus 7 before I updated to 4.3 and still function as expected.
However looking at Google Play tells me that the BBC iPlayer is not compatible with my devices, so I wouldn't have been able to install it had it not been already installed. I seem to recall similar issues on the previous android update, it seems to be down to the developer's over-cautious requirements about OS versioning.
If you haven't got it installed I'm afraid you'll probably have to wait for the Beeb to get their corporate finger out and update the app's compatibility requirements.