Hi G, welcome to the Green! I have a 16 month DD and think it's the best suburb in the world for this age. So so much to do!
In terms of nurseries, sounds like you know the main ones around here. In terms of half days, you could try Pippa Popins. It's uber expensive but just lovely. I know some people who use Treetops and really like it, if that's any help.
Okay, so many activites around here! There is Monkey Music, Amanada's Action Kids on a Tuesday and Wedneday, See-Saw Play Group on a Monday and Wedneday, the Emporio Sao Paola Play Cafe, Gambados, Eddie Katz at Putney and Cupcake Mum just opened.
Outdoors, we are spoilt for choice with the parks. There is Bishops Park (although the playground is under re-construction), Eels Brook Common and South Park.
Plus when the weather gets nice, the White Horse pub does take away pints so the Dads can take the little monkeys to the green and bond 
Hope this helps! If you need links to anything just let me know. I am being lazy and haven't done any.