I was also there on a less-than-24-hour on the way back from TLV. I don't know if you have kids with you, but without it will be much easier.
I arrived and left at night, so I got to the hotel, went to sleep, and had the whole day the next day to go around. I took a bus tour of the city (one of the hop on/hop off ones, I believe it was The Original Tour). I got off to see the London Eye (not worth it) and the Tower of London (worth it). If I could do it again I'd stop at the London Dungeon which I've heard is really good. I was on the bus (and at the attractions) from 8:00am to 3:00pm. I had plenty of time and did not feel rushed.
Check availability for C&P and Le Meridian Piccadilly. It makes getting to the city super easy, cheap, and quick from LHR. Also, the hotel can probably hold your luggage for you if you need them to either before you check in or after you check out.