I've done it and would never do it again - we had a flight time same as yours so travelled down in the evening and set up camp in a public lounge for a very early check in. Didn't get a wink of sleep and it was the most uncomfortable night. I was amazed to find Heathrow semi shuts down overnight but that's when the cleaners move in with vacuum cleaners and the like. Peaceful is not the word that springs to mind! We vowed never to do it again.
The next time we had a similarly early flight from Heathrow we did book the Yotel. Those who believe in karma will be pleased to note that the Yotel was closed due to a plumbing issue, and we were upgraded to a very large comfortable room at the nearby Hilton Hotel at no extra cost. A filthy job but someone had to do it!
Shell out for a short night at a hotel - there are plenty of reasonably priced ones near Heathrow (find one near your departure terminal via booking.com or similar) with shuttles, free buses if they are running, or, push come to shove, a taxi. Adds cost of course, but worth it.
Hope all goes well!
Ian