There's a place on Northcote road (or was) called Boiled Egg and soldiers which looked very child-friendly, but there are loads of places down there and BlondeinLondon says. Or Battersea Rise - there is a Giraffe restaurant there I think.
The cafe on Clapham Common is brilliant. You can sit outside or in. There will be breastfeeding mums everywhere and there are also toys and books for little ones outside. The outside area is fenced off so you can relax, bf in the sun, and watch your toddler play.