I second Science Museum. I took a 2 year old and a 5 year old and both were awestruck.

The newly re opened National Army Museum is good for that age group. There is a play zone, with one hour payable slots as well as lots of interactive exhibits, nice, spacious cafe too and short walk from Sloane Square. My four year old loved it!!

Also Aquarium on the South Bank is always good.

Dinos at the Natural History Museum, and the science museum ... hours of entertainment!

Or if you want to stay closer to home, Bertie and Boo in Balham is always our rainy day go to! X

The London Aquarium went down well with a 3 year old - and I enjoyed it too. But there will be a queue, so best to get tickets in advance.
Also, try the Polka. They may still have tickets + there is a play-space, book corner etc in the large room where everyone gathers before the performance. We ate at Made in Italy, just down the road, at the suggestion of another poster, and it made for a good day out.

The London Transport Museum in Covent Garden is fantastic. Kids are free, you can get 241 tickets for the adults on this site, if you go by train: https://www.daysoutguide.co.uk/2for1-london

Mudlarks at The Museum of Dockland passes some time, and you can take the riverboat them from eg the London Eye (use Oyster - pretty sure it's cheaper with PAYG).

Also the Science Museum and Natural History Museum are fantastic for this age (and all ages!) Get the 345/49 to South Ken!

I know the build it workshop big build weekend workshop is on this Saturday in Chiswick. It's an indoor event.

www.thebuilditworkshop.com/bigbuild

Other than that there's kidszania at Westfield

We are visiting from NYC and have our infant son and 3 year old with us. We used to live in Battersea so thought it would be fun to rent an apartment here but realised we are out of the loop with ideas of what to do with our son. Today we rode the bus and went on the underground much to his delight so wanting some fun suggestions of things he can't do in NY, ideally indoors with this rain.