The one o'clock clubs in Coronation Gardens or King Georges can be rented out for birthdays.
Not cheap though.
Otherwise, the Southfields Baptist Church is always a good option. They have a playgroup there, so there are toys etc that can be used.

Hi,
I think Earlsfield Library rents out the space in the back where they have classes.
Also its a Kids Thing does parties.

Dear Mums and locals, does anyone have any good ideas for an Earlsfield or Southfields venue to have a playtime + teatime for a party for about 10 3 year olds? Ideally something like a playgroup that was happy to lend their venue to parents for saturday afternoons? Thanks