Winter Wonderland in Hyde Park can be fun but crowded and fairly expensive. If you plan on going and want to do stuff like ice skating then book in advance. If you're looking for a Christmassy feel in general then the Southbank Centre has good stuff going on e.g. Christmas market, pop-ups etc. It's worth a look if you're in the area. There's a small market in Leicester Square, which places you well if you want to see stuff like the massive tree in Trafalgar Square, mooch around Chinatown, or head over to Covent Garden to see the shops and displays there. This is the more obvious stuff but it's worth a look.
If you want ice skating then there are a few options around town. Somerset House is a lovely setting. There's a decent rink at Canary Wharf, if you're in the area.
If window shopping is your thing then Fortnum's is worth a look, and places like Harrods. I was there (Knightsbridge) last weekend and Harrods is nice enough, but to be honest I'd only head there if I wanted to shop too. Carnaby St has already been mentioned and is worth checking out, not least because one of the nicest shops in London is at the top of it - Liberty - and is well worth a mooch as it's such a lovely experience.
Pantomime ('panto') is a thing, but more for kids. There are a number of 'alternative' pantos and Christmas shows, which can be excellent. One of the pub theatres near me -
kingsheadtheatre - has something on. I'd recommend one of these types of show in general, if you've not been to a small 'back of the pub' theatre, as the experience is great.
As for seasonal food and drink, the main seasonal food is stuff like mince pies, available absolutely everywhere. And Christmas pudding. For drinking, if you want to be British then just go for quantity
. The mantra "because it's Christmas" can take on an extra-special meaning for having that 4th/5th/6th glass of whatever..
Enjoy!