Hi,
I'm also in Clapham South and can recommend Oaktree Nursery in Ramsden Road, Balham. My son has been there for the last 2 years. It's only mornings, 5 days, so mainly for non-working Mums. It was rated Outstanding by Ofsted and the staff are fantastic. I love it.
The Avalon pub on Bedford Hill has a big, nice beer garden and we take our kids there all the time.
We are also patients at Bedford Hill Family Practice and I agree with comments made earlier.
Also on at The Ascension Church in Malwood Rd is a drop-in music class for pre-schoolers called Fruity Tunes. It's fun and not too hectic. £6 a session.
Glow Lounge does good coffee - Clapham South next to Gigalum. They have an outside space, not great for toddlers but good when you need a good coffee!
Good luck settling in!