I have a 4 hour layover at Heathrow in a few weeks and then a 6 hour layover in Rome (on the same day). Will I have time to leave either airport? (I hear Heathrow is close to the city)
For Heathrow, I'd say not worth risking it but it is workable. You can get to Central London by taxi/tube/bus/special train. The first 3 take ~45 min (taxi cost me $70), but the special train can get you there in 20 min. Use this option to get you to Paddington Station where Oxford St and the like are within walking distance. Take comfortable walking shoes so you can get places quick and not risk being late you get back to the airport.
Yeah, it's doable... Rome more so than Heathrow... But what will you do with your luggage? I hope you're traveling light!
Put the carry on in the locker. If you're in the smaller Rome airport (Fiorino?) than Ryan flies into, it's two bus rides. But getting to the big DaVinci airport is a lot easier and doable. Heathrow probably isn't worth the money to leave and go back. It's not cheap.