if your going to be staying inlondon for a while i recomend biting the bullet and taking the express into town, i know its expensive but its 15 mins vs over an hour by normal time.
Also when i was there i had a reasonably early flight, so i wanted to stay somewhere near paddington (remeber that u can check in for some flights at the Paddington, check your bags and do you last day of sightseeing unencombered)(but check with your airline first)
Anywho i stayed at the atlantic hotel it was 30 quid a room, which was about the cheapest icould find in that area and its about 2 blocks form Paddington, so its easy to walk there. (i managed it with a 35kg suitcade and a full backpack.so anyone can)
urozm8, and even though it may be 1 am according to your body clock you should still be ok for a few hours, take some sleeping pills on the plane (thelights are normally our for about 6 hours on that flight) so u should be able to get some sleep, just hope u dont have a snorere just behind your left ear like i did and u should be fine.