Yep, of course the OP could trawl through old posts, but thinking about this afresh is half the fun
OP I'm assuming your flight lands at 9.45am. Without luggage to collect you'll be landside by 10.30am latest. If I were you I'd forget the HEX and jump on the tube (for a fraction of the cost).
With about 4 hours to play with you want to plan to do more than 'mooch' followed by lunch (there's only so much mooching one can stand).
So, I'd plan to do one of two things:
If you want to take in a museum, some of the best around are in South Kensington, so get off there (40-ish minute direct tube journey from Heathrow). I'd particularly recommend the V&A. Take a look online for their exhibitions.
Or, if shopping is more your thing then get off one stop further, at Knightsbridge, and enjoy Harrods/Harvey Nichols and the shops around there.
After either of those you have multiple options for a half-decent lunch afterwards. Plan to be back on the tube for 3pm and by 4pm you'll just about be airside in T3 - allowing you a good half hour in the lounge.
The reason I think these options are good is the travel itself is cheap, and is also plentiful, i.e. you never need to worry about timings as tubes run all the time. And you get to see a nice part of town and 'make something' of the time you have.
Enjoy