Not to play, but giraffe is a very child friendly restaurant chain ( crayons/ pictures/ little toy giraffe, balloon etc, and nice menu for brunch both adult and child friendly). The one by high st ken is good if you want to let toddler run first as next to Hyde park.
Run first in Hyde park is maybe a good plan. It would just be really nice to catch up properly with friends rather than wolfing food down to ensure I am ready to leave before toddler starts going crazy from being confined too long!
If you don't want a 'posh' lunch I would recommend going to the South Bank Centre - specifically the Royal Festival Hall. Get sandwiches from one of the lovely places nearby, e.g. Pain Quotidien (or the RFH's cafe) and go and eat them inside the Clore Ballroom part of the RFH or around the edges of it. There's loads of spaces for the kids to play and IIRC they sometimes put toys out on the floor of the Ballroom on a weekend. On Fridays there is live music at 1pm. And there are sometimes other activities going on. We have done this LOADS and there are always other families doing the same thing and really no one minds (or seems to) the crazy toddler running around thing at all. It's ace.
there's a great pub in Herne Hill (probably not central enough though) called the Florence - has a whole playroom in the garden, with a glass wall so if your kids are old enough you can abandon them in there & sit outside with a drink.