Kingston to Hampton Court on the far side of the river along the park or indeed, thru the park is most choice, the Surbiton /Kingston side of the river has lots of obstructions alas, though Kingston to Surbiton on that side since the revamp is really rather good.
I can highly recommend the Mute Swan at Hampton Court (near the bridge), a most excellent selection of ales and food (not cheap but so quality).
But in the area I can not recommend enough Richmond park, more so Isabella plantation around april/may when the flowers come out - totally amazing and will be an experience you will cherish, even if you not into flowers, it is mind blowing.
Many good walks in the area, many get overlooked and gets down to playing with google maps and going, that looks interesting. I found a lovely walk along the https://en.wikipedia.org/wiki/Hogsmill_River
that runs from Chessington, towards Tolworth and then along to Ewell to the duck ponds - https://goo.gl/maps/khoiQJoaPyS2
Plenty of plums to be had along that route, as well as the odd apple and pear tree, did find one hazelnut tree but the abundance of grey squirrels soon graze that alas.
Also of note would be the plethora of damson bushes near Tolworth Georges fields, which I made some damson liqueur with this year that proved most potent and enjoyable.
My only real gripe about the London area (least the part I live and frequent) is the lack of Hazel and Walnut tree's, lots of Beach trees and whilst their nuts are nice, they are just not the same.
Though if you are into Blackberry's, the area is dominated with them and can happily graze from mid Summer all the way thru to the end of Autumn.