Hi there, I blog about London for kids and share lots of outings over on my Instagram (museummum.com). Have a scroll through and you might find some ideas - mine are older now, but lots of parents with under 5s use and rate my suggestions.
When are you looking for? Currently all museums and indoor attractions are closed until at least 17 May 2021. After that there is a lot more choice.
London Transport Museum has a lovely soft play area and lots of simulators.
Museum of Childhood is closed for a number of years for refurbishment.
Science Museum has a great basement play area and also Pattern Pod, both free but need pre-booking.
Natural History Museum is great to visit recently due to limited visitor numbers. The animatronic dinosaur and earthquake simulators are always a hit. I've written a blog post on visiting museummum.com/2021/03/19/natural-history-museum-london/
The Postal Museum with a ride on Mail Rail. Their soft play area is excellent but no date announced for reopening yet.
Museum of London Docklands has just refurbished its play area, free but again need to pre-book. Canary Wharf usually has lots going on e.g. Crossorelle Roof Gardens, a little colourful installation in a gorgeous and quiet rooftop garden
Play fountains: More London and Somerset House both on at the minute. Granary Square currently off but great when they're working.
Playgrounds: Princess Diana, Holland Park, Corams Field a bit tired now but usually enough for a little one. Kew Gardens (book ahead!)
Transport: A ride on the cable car, front of the DLR (no driver!), boat along the Thames.
Anyway, here's some starter ideas - please do follow along on my Instagram for more, and feel free to message me there.