Originally Posted by Woodbinerich

I agree, it could end up being a bit tight.
This is my neck of the woods, so forgive the additional advice,
If Fortnums wont budge on the time, you could try an alternative tea spot nearby (better actually - IMO the best in London and an interesting building) like the Lanesborough at 3pm to make things less rushed. It's also marginally closer to Heathrow, without taking your friend too far from the original idea.
If you end up sticking to the plan, and want to go in slightly earlier, the Royal Academy is just opposite Fortnums. The annual summer exhibition is on and defo worth a visit.
Also the shopping around there (Jermyn Street, the arcades etc) is nice. Make sure you check out Paxton and Whitfield for the cheese. There's a long running thread on transporting cheese by air here.