I used to go to White Lion for lunch quite often, remember their desserts especially gooseberry fool in season.
I think the Bulls Head is the one where they have a very good fete once a year with Pig Roast etc, If anyone knows different please correct me.
I have only been to the Engine once and can't remember what is was like. Its adjacent to the station.
There is a good circular walk from Hampton around the lake at Barston and you can pop into the clubhouse at the Golf Course on the Lake.