Cook the ham on Christmas Eve.
Make the soup ahead of time as well - Christmas Eve/23rd for fridge, or earlier and freeze.
Agree about parboiling potatoes early or even on 24th and allowing to cool before roasting.
Roast the turkey. Take it out up to 2 hours before you plan to eat (75 to 90 minutes is plenty). Cover it entirely with a layer of foil and then a large bath towel doubled over to keep the heat in. Make sure it is sitting on a tray that has space for any juices that seep out.
Turn oven up high and heat the fat for roasties. Baste potatoes, throw into the oven for 45-60 minutes.
At the appropriate time, add in your roasted root veg - there are recipes that require steaming/parboiling before roasting, and others that you give a longer time but put straight into the oven. You could use the ring for soup (see below) for parboiling/steaming purposes as they need to be in the oven by the time you want to be heating the soup.
I would also do a pot of mashed potato for large groups, so that needs 1 ring.
You probably also need a ring to steam and then toss in a pan with bacon lardons (and chestnuts if your family likes them) your brussels sprouts.
And 1 to reheat the soup. You can either make the gravy before heating the soup, or make that ahead of time also and reheat while enjoying soup. (Either way, you'll probably want to give it a quick blast just before serving).
If you want another veg, I have 2 ideas.
A cauliflower cheese - yes it takes your last ring to boil/steam, but you can have the cheese sauce made earlier (or steam before cooking mash to give you that spare ring for sauce) but it is very nice if you get it into the oven (15 minutes is plenty) or under the grill (about 5 minutes?) to brown it up before serving.
Frozen petits pois peas - pour them straight from freezer into a heatproof bowl/dish in the morning to allow them to thaw. Pour off any runoff water. Boil a kettle and pour that over the peas as you come back to the kitchen from soup to start plating up mains. Ignore for about 5 minutes. (If you really want to make sure they're piping hot, boil a second kettle at that point, drain 1st off, pour 2nd over). Pour off the water as you are ready to serve and they are perfectly cooked.
I also do baby spinach leaves by pouring a full kettle of boiling water over them draining into the sink in a colander - they are perfectly wilted by that amount of heat. But that may not be special enough for Christmas.
IF you have time and oven space, while eating soup, put your plates/serving dishes into the oven to warm. If not, put them into the washing up basin filled with hot water (and no washup liquid), and they will be plenty warm when you come back - just quickly (assign someone to) dry off to serve onto. I usually fill gravy jugs with boiling water to heat them before filling with gravy.