The borough of Wandsworth is over 75% white. I would be amazed if Wandsworth town had more than 30% of non-white people. It is a very "white" area for south London.
The borough of Wandsworth is misleading - it includes Putney which is very middle class (and white), and hence Putney is in the peculiar position of having very low council tax (originally based on Wandsworth).
Wandsworth Town (which is a small part of actual Wandsworth) is white, and middle class, but centres around a bizarrely small area - Old York Road and the Tonsleys.
The rest of Wandsworth, which is much bigger, think the area all around the high street stretching a bit in all directions, is much more black and some minorities, and is the polar opposite to Wandsworth Town. The area around southside shopping centre is not safe, and is predominantly black.
Looking at figures for Wandsworth can be misleading, as the borough (and areas within actual Wandsworth) can never be analysed properly.