Hi there, I am at the top of earlsfield road so not too far from Southfields. To be honest, it will prob be easier to find out about nurseries and playgroups once you arrive, there are lots of people with kids in this area so often best to go by word of mouth.
My only suggestions are: Earlsfield library - has its own kids bit and they do storytime regularly, plus a few other groups - good place to pick up info on what's on for kids, and local activities etc.
There is an activity place called It's a Kids THing - useful thru the summer. They have a cafe, soft play etc. It's opposite the library.
The Cannons gym on Burr Road offers childcare and is another good place to meet people/find activities for kids.
There are many, many music classes in the SW, you just need to find one you like!
Common Grounds on Wandsworth Common is another popular meeting spot, playground just next to it, they have loads of good flyers for keep fit, music, jobs, sitters etc.
Best of luck, hope you have an easy move!