Don't get a black cab from Heathrow, call AAA on 01628 624446 and get them to meet you, costs about £22 from terminal 1,2,3 and £25 from 4.Also worth phoning the Fat Duck on the day to check cancellations.
Take a cab out to Bray and eat at the Hinds Head pub. It's owned by Hester Blumenthal and the food is wonderful. It's near Windsor - another alternative. There is really not much else very close to LHR... or take the Tube into London.