Day 1: natural history + science museum - definitely book the wonderland in the science museum, it is very much worth it. Don’t eat in the museum cafe because it is expensive and not very nice - we either take packed lunch or go up the road to hyde park if the weather is nice. If two museums in one day is too much for you then on the other side of Hyde park from the museums is Frameless, which my kids really enjoyed (4 and 6).
Day 2: Greenwich - maritime museum, Greenwich park playground, Cutty Sark, bus to North Greenwich then visit the O2 for food, ride the cable car (you can just do a round trip) or get on the Uber boat and go up the river, which is a great way to see the landmarks if you have people with little legs!
Day 3: The Tate modern often has stuff on for kids in school holidays, and from there you can walk along the river to the Southbank centre, and then a bit further along is the London eye.
Other stuff we have really enjoyed in London - dinosaur afternoon tea at the Ampersand hotel; visiting Hamleys; there are often kids performances in West end theatres (but be aware it tends to be aimed at either preschool, or 7+, so might be hard to find something you all like. The postal museum (sounds boring, but there is an underground train! And its play area, ‘Sorted,’ is great, have to prebook).
Stuff that doesn’t work for us: walking or bus sightseeing (Big Ben/Buckingham palace/Trafalgar square) - the ‘interesting’ things are too far away/interspersed with waiting. Merlin type attractions (Sea life, London dungeon, Shrek adventure) - very overpriced and don’t last that long. We haven’t done the Paddington experience but I’m suspicious it would fall into this category. London transport museum - much less child friendly than you think it will be, and really expensive.
Can’t help with places to stay because I live here but generally you could stay very central and it be very convenient but expensive, or slightly further out - if you choose the further out option then check the travel time because some tube lines are great (eg elisabeth line) and some horrifically slow and unpleasant (eg South end of the Northern line).