Definately Merton abbey mills is the most popular place for barbel. However they are all through the river it's just some areas are just to dodgy to be fishing on your own. Me and Sarto tried the section before Wandsworth and although we had no trouble that day it didn't feel very safe and in the end we spotted a gun in the river and left.
For all thefishing I have done between carshalton and morden hall park I have had chub to 4lb, trout to 4.8lb, carp to 6lb plus good size roach, dace, gudgeon, eels but only two tiny barbel which were more like gudgeon. They are there but they are difficult to catch unless you just get lucky. The best place is morden hall park angling clubs stretch but you will need to join first. You can get a form from the information office in the park. Also they have a website with loads of info on and a forum. However they can be slow to respond to new posts on there. I moved away from the area in December and miss it a little, will fish it again one day though, it always throws up a surprise for me. There are several reports on here if you do a search for them. Best of luck, your really gonna need it!!