Every age, many times!
1-2 yo
London aquarium
Bach to Baby
Chase pigeons in Trafalgar square, and pop into an art gallery for them to have a quick peek and enjoy the space!
3-4 yo:
(take scooter rather than buggy)
science museum ('garden' hands-on area in basement)
transport museum
Battersea Park zoo
Diana Memorial playground
boat trip on the Thames
Brilliant childrens theatre at Unicorn, Polka, Little Angel, My First Ballet at Sadlers Wells
5-6yo
(top floor of buses and let them go through tube barriers themselves!)
Dinosaurs at the science museum
South Bank (buskers, carousel, the river and the London eye. Also a playground)
Hamleys/Harrods
Changing of the guard, Buckingham palace (outside), Trafalgar square
Tower of London
7-8yo
British museum
London zoo
Greenwich museums
London dungeon
The Shard
9-10
Houses of parliament
Cabinet war rooms
The Old operating theatre and herb garret
Thames RIB trip on Thames (speedboat)
Art galleries, children's concerts and theatre, courses and experiences
Older
Everything an adult would enjoy! Historic sights, shows, music, art, food, shops...