I'd echo the London souvenirs idea.

London is definitely open around New Year. I think it's a great time to be in the capital. You should plan for New Year's Eve itself though, e.g. tickets to see the fireworks if that's what you want, or dinner, or whatever else it is you have planned.

The New Year's Day parade can be fun to watch, and the atmosphere's nice. As it's also through the centre of town you're well placed to shop after if you want to. The shops will be open. British Museum aside, some of the best museums are open on New Year's Day. I'm thinking the Natural History Museum, and the V&A in this context in particular.