Try bushcraftparties (google it)
We used this company for a party (9 year olds but they adapt for any age) on the common last summer - they were BRILLIANT!! A lady from Clapham ran ours (Joetta) and the children loved it and her.

The Spinney! They have a great mud club and you can hire the venue exclusively. We had an amazing 4th birthday party there end Sept. Message them through IG spinnethe

Have a look at Oasis nature garden - they have a beautiful venue just behind Stockwell station and do parties at the weekends.

Do you know of any party planners / organizers who will do a forest school type birthday party at Wandsworth Common, Clapham Common or Battersea Park? We are trying to plan an outdoor birthday party in December for 4-5 year olds.