Just some old news John which I thought worth a mention, many years ago saw some big carp just off a boat slipway at Twickenham. About 5 years ago walked the Richmond stretch and a bloke pulled in a 7lb barbel. I think he had been fishing quite far out with a swimfeeder but, I cannot remember what he had on the hook. It was where the river bends halfway between Richmond and Twickenham where there's a big green on the Richmond Park side; about beneath the old Star and Garter home on the hill..
Thirdly, Last year I spotted a punt hire place at Hampton Court, 50 yards up stream from the bridge, opposite side to Hampton Court (I think it was about £28, I did have a card somewhere but, cannot find it now)---thinking of fishing the weirs there which I believe hold some good barbel. What the license, rules are about that I am not up to date though. Maybe have to check that out for yourself; however, looked very nice fishing if you can get to the weir which you can only do by punt and not too far from Richmond.