I use LHR a lot. 4 hours between flights (from landing to take-off) is not enough time to see London.
Flights from the US tend to come in at about the same time so the immigration line is frequently very long and can be very slow moving. I noticed during my most recent trip that the questioning is now taking longer than the usual <5 minutes.
Factor in about 45-60 minutes for deplaning/immigration. You can be into Paddington in about 25 minutes via HEX if you don't have to wait for the train. Then via Tube to tourist sites, add 20 minutes. So this is about 1 h 45 min already.

Last time I was at LHR, it took me about ONE HOUR to get thru security to get to my departure gate! And this was in late April during a non-busy time (no school holidays/bank holidays). The lines were monstrous!

This is nearly 3 hours WITHOUT factoring in the time it will take to get back to LHR (30-45 minutes depending on how far you are from Paddington). Factoring this in, you will not even get to your gate on time for your flight.

Remember you need to be at the gate at least 30 minutes before take-off. Taking all this into account, you won't have ANY time to see London.

London is such a wonderful city and merits a trip all to itself. A whistle stop tour, esp if you are sleep-deprived, would not do it justice.

Why not just go onto the next terminal for your Rome flight for some shopping and even a catnap?

If you want to get out of the airport, consider Windsor, much closer.

listen to janis. i fly in and out of heathrow about twice a week. 4 hours from arrival to next flight's takeoff is not enough to go into town. she mentions all the little snags that might happen but you also need to consider the fact that you don't know where you are going (i assume) and it takes some extra time just to figure things out and find things...in the airport, the trains, the taxis, or whatever. the time would be so tight that even someone who knows exactly where he was going would have a challenge. someone unsure of what they are doing does not have a chance.

and to further janis' point...things always go wrong in london. the trains regularly have delays, there is always "unexpected" traffic, etc. if i absolutely have to be somewhere in london (e.g across town), i will usually double my travel time in order that i won't be caught out by a problem.

many people here quote travel times that are not realistic under normal conditions and for people who don't know where they are going. these are hypothetical times that can only be realised under the best of conditions by someone who knows exactly where he is going...and who moves fast.

I just thought 'No' as an answer to your question first off. The others have given you details as to why not. Everyone knows how awful CDG is. To me, LHR is about the same. Four hours is not a long time to see the city and be back in time to catch your flight...

So, islandmom, you're being told to completely ignore the fact that you're a fifteen minute train ride from one of the most interesting cities in the world - regardless of whether you arrive on time, breeze through I & C, can follow directions explicitly and get it right 100% of the time, and don't experience the slightest inconvenience from jet lag.

I say make up your own mind on the spot.