If you just want to clean up etc after the flight - I would say your best bet is to go to the AC Arrivals Lounge in T3, and hang there for a while. I dunnow if they will take an MLL pass (I presume so) but when I was last there is was pretty quiet (only 2 of us in there) so I am sure if you were to grovel enough she would let you in. Even so, you would still probably have a lot of time to kill, and it is small/not much to do there, but at least you can shower and get some 'down time'. If you have an evening connection and a bit of coin - I'd get a day room. I usually go for the T4 Hilton, which is attached to the airport and hence easy access to tube and LHR Express train.