By age, what is free, and how to get around with a pushchair.

The short version: London is a superb city for children if you plan around naps, distances and the Tube's lack of lifts. Pick one thing per half day, keep the parks in reserve for meltdowns, and let a car do the long hops.
All the national museums, all the Royal Parks, the Changing of the Guard (11:00, check days), the Southbank skate park and fountains, the view from the Sky Garden (book), the Museum of London Docklands, and Greenwich Park with the Meridian line. See London on a budget.
Under-11s are free on the Tube and buses, which is excellent until you meet a station with 60 steps and no lift. Buses are the friendlier option (the top deck of a number 11 or 15 is the cheapest sightseeing tour in London). For hotel to museum with a pushchair, a fixed-fare car is about £25 across the centre, and child seats are free. For Heathrow arrivals with two children and four cases, a 6-seat MPV meets you in the terminal at a fixed price with 60 minutes free waiting.
The food halls (Market Halls, Arcade) solve the problem of four people wanting four things. Pizza Express, Wagamama, Franco Manca and Dishoom (breakfast is the calm sitting) are the reliable chains. Afternoon tea for children runs at Fortnum's and the Ampersand hotel with a science theme. Most pubs allow children until 21:00 and do a Sunday roast that a seven-year-old will eat.
The Natural History Museum dinosaurs, the Science Museum, the Diana Memorial Playground in Kensington Gardens, the Changing of the Guard, the pelicans in St James's Park, the Museum of London Docklands' Mudlarks gallery and the whole of the South Bank.
From about five, when museums and the parks land; from eight for the Tower, the Studio Tour and the theatre. Under-fives get the parks, the zoo and the boats.
The parks and museums are; the Tube is not (many stations have no lift). For anything more than two stops with a pushchair and bags, a fixed-fare car with a free child seat is the sane option.
Under-11s travel free on the Tube and buses with an adult; 11 to 15 pay half with a Zip card or a visitor discount set on an adult Oyster at a station.
Terminal confirmed from your flight number, pickup timed backwards from check-in or tracked forward from landing, name board standing where you exit — London's entire airport system run the way it should have always worked. Sixty seconds books any of it.
Visitor guides: Best time to visit London · Things to do in London in winter · London airports guide
Further reading from our Medium blog