London with kids in December, by train: our only trip to the United Kingdom, and it was good, within limits. We slept just outside the centre, took the bus everywhere (top deck, which is half the outing) and did the Harry Potter Studio Tour: four hours, and it could have been longer. Hamleys, six floors of it, to finish.
Is London expensive with kids? Yes. The Christmas markets in the centre charge eye-watering prices, a waffle or crêpe most of all. Budget for it and it is perfectly doable. Everything is in Christmas in London with kids.