London in December will take every penny you offer it, so offer fewer. The lights are free and so are the big museums. The hotel bill does the damage. Rates tend to climb as Christmas gets closer and jump again for New Year's Eve, so come in the first fortnight, when the decorations are fully up and the office parties have yet to claim every pub in Zone 1.
The West End lights normally go up in mid-November and stay into early January, and one evening on foot covers most of them, from Oxford Street and the angels over Regent Street through Carnaby and Piccadilly down to Covent Garden. Visit London publishes a free map of the route. Finish in Trafalgar Square. Oslo has sent the Christmas tree there every year since 1947, in thanks for Britain's support during the war, and in most years it is lit, with carols, on the first Thursday of December.
With about 3.4 hours of sunshine a day and rain on around 12 days, the afternoons want a roof, and the national collections supply one on the taxpayer. The British Museum, the National Gallery, Tate Modern and the three big museums in South Kensington all let you into their permanent collections for nothing; only special exhibitions charge. Save the Tower for a dry day. Pack a proper waterproof and gloves and leave the umbrella behind. The wind off the Thames will have it inside out on the first bridge, and highs of around 9°C feel a good deal colder in that wind than they look on paper.
Hyde Park Winter Wonderland is free to walk into at off-peak times, though you still have to book a slot, and once you are through the gate the rides and food stalls get their money back soon enough. Walk round with your hands in your pockets and leave. Skate at Somerset House, in the courtyard beneath a very tall tree, is the handsome rink and charges like it. Weekend sessions cost more, so book a weekday.
For theatre, walk to the Official London Theatre Ticket Booth in Leicester Square, run by the Society of London Theatre and home of TKTS. It sells discounted and half-price seats for the same day and for up to a week ahead. Arrive with a shortlist of three or four shows and let the board decide for you.
The New Year's Eve fireworks on the Thames are ticketed. Without a ticket you will not get into the viewing areas along the Embankment and around the London Eye, and none are sold on the night. On Christmas Day nothing runs at all: no Tube, no buses, no trains, not even the river boats. Boxing Day manages a reduced service. If you are in town for Christmas, book a bed within walking distance of wherever you mean to eat lunch on the 25th, or set the cab fare aside before you spend anything else.