We’re National Trust members (you can get a very very good deal via Quidco) and often visit Wakehurst Place – for free entrance you can be a National Trust or Kew member.
For this you get acres of green to walk around (and hills too, there’s some very buggy unfriendly ones, but we did it!), lots of plants, flowers and trees.
We’d not spotted them before, but there’s a few areas which are marked with an asterisk (on the map) which are designed for younger visitors, H being almost two enjoyed two of the ones we went to.
There’s also two cafe’s, both very child-friendly. There’s changing, and the whole area is pretty much safe for a child to run around (beware, there are ponds and a lake though), as well as plenty of ducks.
The shop at the end is also good space wise to get your buggy round!