In late April, 2012 my wife and I had to do a similar trek at Heathrow. Let me give you my experience and you can decide.
1) Arrived at Heathrow Terminal 3 on an American Airlines flight from Chicago (U.S. Passport) at about 9:00 a.m. local time. Walked to passport control and waited in line until about 10:30 a.m.
2) Collected our bags which were already waiting for us, had nothing to declare so breezed out of Terminal 3 to catch Heathrow Express train to Terminal 5.
3) Arrived at Terminal 5 about 11:10 a.m. and rechecked bags for flight to Paris. Finished this part at about 11:30 a.m.
4) Had our passports checked again and went back through security at T5 quickly and arrived at T5 waiting area at 11:45 a.m.
5) Gate posted for 12:40 p.m. flight at 12:10 p.m. Fortunately only a 5 minute walk to gate from where we were at.
Bottom line - From landing at T3 to get to T5 waiting area took a total of 2 3/4 hours.
Heathrow is not as easy to sleep in as for example Stansted. I looked for a nice place to sleep in Terminal 3 but decided that this was not a real option. Benches are almost only available in the first floor, in a corner far to the left close to a pub. Of these, only a couple are without armrests and upholstered. As of 2008, it was not recommendable to sleep in the ground floor (arrival and check-in area) as construction works are taking place. And never try to sleep on the carpet with is not comfy, just dirty! Shops, including cafés, are not open during the night.
Terminal 4 is a little different, you’ll find a couple of scattered benches with a group of them in a waiting zone close to the BA first class check-in. The same as for Terminal 3 with only a couple of the banks being upholstered and without armrests. Here again, be creative and find out the most comfy way to get some sleep by arranging bags, chairs and baggage trolley. There is a 24h Starbucks café in the arrival area, but no other shops – only a couple of vending machines. Although in sleeping qualities far behind Stansted, terminal 4 is a better choice than terminal 3. It is said that there’s also an acceptable sleeping area in the arrival hall of terminal 2. But in Heathrow, the best choice is to spend as less time as possible anyway…