We recently went for a lovely autumn walk in Richmond Park and parked at Sheen Gate. Since the above picknick option looks tempting but is obviously hardly feasible in real life, we opted for the next best thing and went to the Victoria pub, which is a 10 min stroll away from the gate through quiet streets, situated in the original part of Sheen (church, school and pub all on one street - as it should be!).
The atmosphere is nice and relaxed, the food is good. On weekends you have to book well in advance for lunch and dinner since it seems that most of Sheen can be found there enjoying their roast. It's not cheap, but I think well worth a trip.