I believe London Buses and Underground ran a limited Christmas Day service until about 1980.
I happen to have a 1972 LT buses Christmas booklet to hand -
Christmas day had a limited service on a limited number of routes (not all operating the full length of the weekday route, e.g. 12 only ran Oxford Circus - Norwood Junction, and it was only about hourly south of Dulwich) - broadly, buses didn't start until 0800 or 0900, and all services had finished by 1600. (From memory of seeing it on paper somewhere, the bus crew agreement for Christmas day was single spell / maximum 5 1/2 hour duties, and all buses back in depot by 1600.) This included some of the (then) Sunday only routes, e.g. 180A (New Cross - Abbey Wood)
Night buses did not run on the night of 24/25 or 25/26 December, as now.
The Underground ran most but not all lines, with some stations closed, starting from 'about one hour later than on a Sunday' and finishing at 'approximately as on Sunday' - with reduced frequencies until 1400 and even more reduced freqencies after that.
Boxing day had a reduced version of a Sunday service on both.
I have a faint memory of BR (Southern) suburban trains being a 'reduced Sunday service' on Boxing Day in the late 70s, whiich meant an hourly service rather than half hourly, my parents had not enquired about the detail, and we went for the 'missing' train on a trip to visit relatives in Ealing - 30 minutes wasn't quite long enough for it to be worth going back home and coming out again for the one that was running...
(edit to add - on reflection, the above might just have been a bank holiday on 27 or 28 December, not actual boxing day, if 25 and / or 26 December had been at the weekend.)
New Years Day was (from memory) Sunday service.
LT buses continued to run a reduced service on Boxing Day until fairly recently (as in within the last 10 years or so) when it became Sunday service.
In the late 80s, there were some odd allocations as a few garages / operators were closed for the day, which meant in a few cases, different operators ran routes for the day, e.g. Metrobus having a one day operation on route 208 (photo - not mine - on Flickr
here) A few routes extended beyond normal termini (to cover part of another route that wasn't running) and there were a handful of one-off routes, like Ensign Bus 86B (photo - also not mine - on Flickr
here) which covered parts of other regular routes in the area.