Will you have a car? If so, go visit Deen City Farm, a short drive south, it's free. They do pony rides but I think kids have to be older (3+ maybe).
Also Battersea Park Zoo, pricey but good, easy to reach from CJ. And Vauxhall City Farm, I believe they have horses too.
Softplay at Eddie Katz (Earlsfield) or Bertie and Boo (Balham/Tooting).
Bertie and Boo cafe on Northcote Road has an indoor playroom - good for a rainy day.
Ducks on Wandsworth common (buy duck food at the very child friendly Skylark Cafe in the middle of the common).
Playgrounds on Wandsworth common, there are 3. And there's one on Clapham Common.
There is a drop in playgroup with craft and toys (indoor and outdoor) just off Chivalry road - afternoons and Saturday mornings.
Plenty of other playgroups around, usually in church halls. Try the one at St Marks, but it's very popular so get there early.
Various music classes, some run termly which wouldn't suit you but some offer drop in. Try Amandas Action Club which does drop in.
Have fun!