A long time ago I saw some very big bream come out at Sunbury, you could get the ferry over to the island and there was a weir there that you could just fish downstream from. Whether you can still do that I wouldn't know and it all may have changed now but what the bream liked there may not have. They were caught out in mid river on swim feeders and you may be able to fish from the bank without having to get over to the island. All a bit of a long shot but I thought it worth a mention, could be something to check out.
There were reputed to be barble there as well but I never saw any but I did see one come out at Twickenham a few years back and I once saw a big shoal of big carp at Twickenham as well, near Eel Pie Island. Again, all a long time ago but might wet your appetite, it does mine, love to get back there one day and have a go.
I was exploring the possibility of hiring a punt in the summer at Hampton Court a few years ago, go out and fish the big weir there but never got round to it, maybe I should investigate again. I always found the Thames one of the most intriguing of rivers, I still think there is a lot of unknowns about it.