Just a little update: I asked the garage to diagnose if the issues were caused by a faulty alternator or if the battery needed replacing. Apparently both were fine. The alternator was charging the battery and they encountered nothing that indicated a problem. I obviously drained the battery with my stupid attempts to change the window button on the power unit!
Seems like my alternator and battery were fitted recently, just before I bought the car. The alternator is apparently of a make that is sometimes temperamental so I'll keep an eye on that. Got my car back though so one very happy BMW owner and not having to shell out £400.

Thanks to everyone for their help.

Bottom line: do not blindly trust the AA's diagnosis.