I took my boys to central London quite a lot when they were small. The museums are free and I would recommend the Science Museum and the Natural History Museum, if I recall they are next door to each other.

Also, I agree Oxford street is not what it used to be, the Kings Road area is ok for you, but maybe a bit boring for him. Hamleys and Harrods are good though, Harrods also has a pet department, which I'm sure he'll love.

For a browse round, Covent Garden is great loads of Market stalls, and street entertainers and I'm sure they'll put on some Xmas stalls. You could then follow this up with a child friendly show like Lion King or maybe a Panto, all very close to Trafalgar Square. You could also eat in Planet Holywood which is in Leicester square, my boys used to love it.

A word of warning, some of the tourist attractions like Madame Tusuards and the London Dungeons can be very expensive and my boys were terrified of the Dungeons had to leave 5 mins into the attraction!!

Hope this helps.