The MLL at LHR is OK but nothing special. It gets very crowded in the afternoons when the bulk of the westbound transatlantic flights leave. If you are flying on AA I'm not sure how you would access it, but if you are *G and looking for a quiet place to put your head down you might want to give the SQ SilverKris lounge a try. If it's open it is usually quieter. I'm not aware of any showers in the MLL (in fact I'm 99% certain there aren't any) but there is one in the UA RCC across the hall (ie other side of the moving walkway)