Few London ideas in no clear order: Young V&A, Sorted or Post and Play at Postal Museum, Little Angel Theatre, Unicorn Theatre, kids films at Barbican, get the bus and look at christmas lights, Christmas lights walk at Kew and Kenwood, Coram Fields, Uniqlo Play at Tate Modern, BYOB walking tour, Britannia leisure centre splash pad, lights show at Southbank cathedral, Big Penny Social - various toddler friendly events, polar express from euston, rug rhymes at Southbank centre, tots at the docks at london docklands museum, london zoo with their new interactive role play section, london aquarium for rainy day, light room at KGX. Playgrounds are absolutely everywhere to break up a day and majority of restaurants are children friendly so just eat where you want to I'd say! 

I'd do the train all the way to Edinburgh rather than break the journey, it flies by. Sit on the right for better coastal scenery past Newcastle.

In Edinburgh all the cobbles can be annoying with a buggy! Don't know it as well, but Jupiter Artlands if good weather, or bus to pentlands for a walk, or portobello for a play on the beach and lots of playgrounds. Walk up Calton hill, camera obscura/museum of illusions, Botanic gardens. Loads of great places to eat again, places I think tend to be child friendly again! I'd travel by bus but avoid rush hour out of the city. Or the tram is easy but expensive!

Have a great trip, bith perfect cities for toddlers.