My brother used to live on the Mole at Hampton Court, Molember Road I think it was. It was all privately owned, the houses along there had the fishing rights as I remember rightly and it was gated where my brother lived. And as above they are all pretty wealthy people and will be on you like a rash. if your fishing in the wrong place.
I used to watch a heron leap off the towpath and pull out chub from my brother's house, I was fascinated how he got back on the tow path and I could have got permission to fish from his neighbours but it was always family time, not fishing time.
However, I believe the area known as cigarette island just by Hampton Court Train station edged right up to where the Mole flowed into the Thames, whether there was any actual Mole fishing at that point I never walked that far to find out. Cigarette island is free, so you can fish the Thames there which holds barbel and some carp I believe but there should be other species present, roach, bream etc, maybe Zander as they are caught at Kingston I believe. So, maybe worth a visit and find out whats what, maybe you will find some other anglers who will fill you in but they mainly night fish for the barbel.
There was a tackle shop at Hampton Court, but I think it maybe closed now, my brother knew the bloke who ran it, Bob the Fish he called him, but see if you can check that out, might not be closed.
Maybe worth finding out if there are any clubs in the area as well that might have fishing rights on the Mole.
Sorry it is all a bit second hand info but my brother has moved from there now and it was a few years ago so, I don't go there anymore.
However I always fancied taking a punt which you can hire in the summer and fish the weir just upstream at Hampton Court but never got round to it, flying visits and family ties etc. But that weir always looked very fishy to me, I believe you need some sort of permit but whether you need one from a boat, I dont know, never got that far.
But the Thames along there always looked interesting, fishing near the Lock area for example looked like it might hold some big roach in the slack areas.