Swan's Rest Island and Rivermead Island

One island now and effectively part of the Right bank.
(i.e. it doesn't look as if the Right bank stream could be followed - or if at all only by a canoe)
Kempton Park Race Course is a half mile north, on the Right bank, the other side of the A308
[ I had a note of a slipway - but I can't find one now. ]

Jerome K Jerome -

WE stopped under the willows by Kempton Park, and lunched.  It is a pretty little spot there: a pleasant grass plateau, running along by the water's edge, and overhung by willows...
The selfishness of the riparian proprietor grows with every year.  If these men had their way they would close the river Thames altogether.  They actually do this along the minor tributary streams and in the backwaters.  They drive posts into the bed of the stream, and draw chains across from bank to bank, and nail huge notice-boards on every tree.  The sight of those notice-boards rouses every evil instinct in my nature.  I feel I want to tear each one down, and hammer it over the head of the man who put it up, until I have killed him, and then I would bury him, and put the board up over the grave as a tombstone. 

Above quote is recommended reading for the Environment Agency!