About sleeping on the plane--depends on what time you get into LHR. If it's later in the afternoon, ie: 1:00p and later, I'd suggest getting a good night's sleep the night before your trip, and then staying up the entire flight (unless you have a dinner meeting or something such as that the evening of your arrival).
From my LHR flight (SFO-LHR), we left about 1p SFO time, and all of the cabin lights were kept off for takeoff, and all through the flight (there was sufficent light from the windows for the meal service and galleys), allowing passengers to "naturally" settle in to "sleep mode." The sun sets fairly quickly when flying Eastbound! Of course, I made the HUGE mistake of staying up the entire flight (as if I'd miss something happening over Greenland in the night!), and when we arrived in the next morning at 5:55a, I was paying for it! I had to have a caffeen I.V. in me the entire day!
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