Young V&A is amazing! Horniman has outdoor play, animals, and indoor museum (small aquarium for a small charge). Science museum, London docklands museum has an amazing play space for little ones but it does cost £3pp.
These aren’t all free but I’ve found this list really helpful https://museummum.com/baby-toddler-london-museums/
Tate Britain currently have 3 play areas
In the end we went to the Science museum, the Garden there is a fun play area for children to play with water, buttons, mirrors, giant Lego.. all for free. Recommended!
Would love to hear suggestions