Boxing Day is kind of a neither one thing nor the other day. There's no National Rail trains, so the main arteries in and out of London can get busy, but it's nothing on a working day. The main shopping areas will get busy for the sales (expect to see the usual newspaper articles on how disappointing the sales were - we do enjoy a good moan) but otherwise, it can be quite quiet, as people sleep off the excesses of the previous day. There's kind of a mish-mash of what's open and what's not - best to check individually, as there's not one hard and fast rule.
There's traditionally a load of Boxing Day football matches, too. In London, Fulham, Spurs, Millwall, QPR, AFC Wimbledon, Sutton United and, yes, even Dagenham & Redbridge and Dulwich Hamlet will be playing at home.
The 27th is more of a working day - however, lots of people take the whole week off, so it'll be more like a weekend day.
Things to do - well, if you like shopping, and can cope with bargain-hungry crowds, that's an obvious one. There are ice rinks up all around the city at this time of year (you need to pre-book) if that sounds fun to you. There is a slightly gaudy Winter Wonderland in Hyde Park, and a load of wooden huts selling mulled wine and the like along the river. Kew Gardens and Greenwich have winter festivals on. This Londonist article gives a few suggestions:
https://londonist.com/london/christm...-on-boxing-day
Do be prepared for the weather. It's generally cold and damp around Christmas, although the long-range forecast is showing fairly mild, dry weather in London this year, so you could be lucky. It will cool off quickly when the sun goes down, which is around 4pm.