I roll over a spreadsheet annually for presents and budget - it's easiest.
Lists of jobs to get done and foods to prep and presents and food and other bits to buy.
Like the spare batteries, extra logs for the fire (and matches and firelighters!), plenty of loo roll and bin liners....
Like the annual checking of the smoke alarms (I know, it should be monthly, but better to do it then than not at all), and cleaning of the chimney...
I buy cards as soon as I see them. If I have any reason to travel and be stuck on a train or in an airport/plane, or just sitting in a waiting room for a while, I will bring a few with me to get out of the way. So they are all ready by posting date (at least in theory).
Empty the freezer over autumn to clear out old food, and make space for both preparations and leftovers for Christmas.
Preparations include making a few extra family sized regular dinners during the autumn and freezing those (lasagna, curries, etc) to have in those manic days coming up beforehand (running through for dinner before the nativity, grabbing a bite after a Santa party and before an adult party, unexpected visitors dropping in around dinner time needing a plate...).
And I try to book an internet shop for food most weekends in December because the shops are so manic. Getting enough for the week and most of what I need for Christmas (the heavy things especially).
Things I know I'll want, like PJs for all, I will pick up early. But I only tend to wrap everything in the last couple of weeks - I pick a night (preferably when DH is out at something else), pack DD off to bed, turn on some Christmas music (a movie is too distracting). lay out all my paper, ribbon, tags, scissors, pens and tape dispenser on the table, presents dragged out from cupboards to be all there, and set up an assembly line. I have ordinary paper bags (or fancy Christmas ones if I have them) to be able to split the gifts by family/location once wrapped (saves time - I can neatly pack up each bag later) and allow myself a reward of a nice glass of wine and a good mince pie once I am done.
I tend to prep ahead of time, as work and home life are manic in those last few weeks. But if I had to do it last minute, I would still be doing it the same way - just doing more jobs at once or bigger shopping trips (I tend to try and buy one present a week once the DCs go back to school in September as I have about 50 people to buy for in total).