Checked some of it out last year although never actually fished it. I visit a brother nearby.
I checked hiring a boat to fish the weir just up stream of the bridge. You can do this apparently. I think they were about £28, something like that. A nice little fishing rowing punt. This is hired from a place just up stream from the bridge, about 100yds left hand side looking up stream. However, I think they are only doing business in the summer. There's barbel there and I suspect the usual species so I am told. Anyway, weirs are always interesting.
The other place anglers get is Cigarette Island. Its not a island but it is the bit by the bridge going downstream by Hampton Court Station. I believe there's barbel there as well. I believe its pretty much free fishing both sides , although some of it is not accessible because of houses. The far bit over the bridge going up stream, hence the interest in the fishing punt because this is where the weir is. Sods law.
There's a tackle shop in Hampton court town, you will have to ask the locals where it is, a bit complicated to explain, its not far from the station though, 10 minute walk. Cross over the busy road from the station into the town and turn left sort of. You will be heading in the right direction.
I believe there is fishing in Hampton Court but, not sure and the river Mole runs out at the end of the cigarette island bit, but, I think the Mole is very restricted, the tackle shop will be better able to inform you.