Definitely walk or take a boat along the river. Loads of things to see along the riverside - Westminster, Globe Theatre, Tate Modern, cross the bridge to St Pauls, Golden Hinde, Tower Bridge, Tower of London. Usually various buskers along the river which are fun to watch. To me the river is what feels most like “seeing London”.
You could do the London Eye but I think my DC would get bored, it takes 25 minutes and goes slooooowly (they are a little younger than yours tho). You could consider the Shard or the Sky Garden instead as a way to view London? Sky Garden is free (tho you need to pre book) which means you could abandon it without loss if the weather/view will be rubbish.
Possibly Buckingham Palace and St James’ Park. Not sure if you can actually see Buckingham Palace from any bus route, but there will be plenty that stop quite near.
Natural History and Science Museum - good if the weather is poor - get there as early as possible in the am, and get tickets for Wonderland at the Science Museum, it’s not free but it’s really worth it for kids. Also do book a local restaurant for lunch (plenty of kid friendly options nearby eg Comptoir Libanaise and Pizza Express) as they are always heaving.
Consider a wander around Covent Garden, Long Acre, Seven Dials, Leicester Square, Chinatown. No specific tourist attractions to go and see there (except the transport museum if you fancy it) but it’s a great atmosphere. You could start at Trafalgar Square (charing cross tube), walk via Chinatown and Leicester Square, end up in Covent Garden for an ice cream or similar.
Whereabouts will you be staying? As there are some non central places well worth a visit but perhaps not worth the trek if they are the other side of London. For example Hampton Court/Kew Gardens, Greenwich/Cutty Sark.
I would skip Tussauds and London Zoo personally. I found London Zoo rather sad compared with more spacious zoos outside London.