Trust properties are not always that close - the only ones I can think of are hatchlands and clandon park in Surrey, and the latter is now a burnt out shell. There are also mileage limits for volunteers which makes sense.
Mainly though, people choose a property because the roles there suit them. The next nearest one may not suit.
As I said, no dress code, but I think it only reasonable not to wear anything covered in slogans or logos. Not that I own much like that.
I do wear a poppy at the right time of year, but it isn't compulsory and should not be.