Oh Hamleys yes. Thanks for all the ideas. Great place to start. Is Covent garden still entertaining with street performers? Haven't been there in about 12 years but I remember spending hours meandering there and visiting the transport museum.
Winter Wonderland in Hyde Park is free to get in but the rides and stuff are pretty extortionate. We had a couple of hours there, then walked down Oxford Street, saw the window displays in selfridges, hof and Debenhams, the lights in regent st and carnaby st (my favourites) and ended up at Hamleys. It was all mobbed on a Sunday afternoon though! We just went on one teacup ride and had german sausage (and mulled wine) at winter wonderland so didn't spend loads but it would be v easy to spend more. They have children's shows (some Disney on ice type thing) and a big wheel as well.
South bank is nice for street performers around festival hall / QEH. They sometimes have free concerts in the lobby as well, and there's a lift that sings the floors at you. Tate modern is free and walkable. I think there's a Christmas food market along there at the moment as well. The Thames clipper is fun as well, although not especially festive.
Thanks for the tip about Hyde Park @hobbes8. Good to know its pricey we may avoid. Are there any good playgrounds in the city parks which are good on a cold but dry day? Tate Modern is on the to do list!
For ages 4&7 I'd only do the science museum if you do the paid for bits. The rest of it was really disappointing for our 2 last year at that age- by contrast we had to drag them out of @ Bristol after 6 hours a few months later.