I'm on the other side of Tooting near Balham and my children are way past the baby/ toddler stage but I believe there is a lot on near you and surrounding areas - play groups, baby swim, music classes, exercise classes for mums. I really recommend joining the Tooting Parents Network ( a huge Facebook Group) and I have no doubt you will have lots of friendky people offering to meet up / show you the ropes. Furzedwon is near you as well which is full of families. In many ways this is the best time to move as so many opportunities to make friends with a baby or small child.
Best wishes and good luck with the house move!