Jan 1 lunch/early dinner near South Bank, London
Many, many many options near P. Circ. However, I doubt if many London Hounds carry lists of Jan 01 opening hours in their heads.
Anyway, here's a good place to get you going:
Also Google: Wilton's London, Ritz London, Greenhouse London (short cab ride), Le Caprice London. All good.
You'll need to make your own inquiries about 01 Jan. opening.