Short answer: badly!
Long answer: do different bits badly at a time. So, when ours aren’t sleeping, the priority is sleep and survival and doing the bare minimum to keep afloat (school uniform, personal hygiene, stuff like that), and accepting that this too shall pass and even though I haaaaaaate having piles of clean washing in our bedroom that we have no time to put away and we’re all tootling through the bags to find clothes, at least my hair is clean and I had a lie-in at DP’s expense after I did a night with the toddler.
We alternate nights on the monitor with the toddler. Other grownup takes the primary-aged one who sometimes has nightmares or wets the bed, but generally sleeps through.
Not everything gets done! When we prioritise X, Y drops off the radar. We had a cleaner for a while but can no longer afford it and tbh, though the house is now filthy at least I’m not stressed running around tidying it while getting two kids ready to get out the door (DP commutes so mornings are on me).
Meal plans, batch cook, divide chores so one person does all of one task so you’re not constantly doing “handovers” or thinking the other person should do it. DP does laundry, organises the food shop, bins, a load of other things. I do meal plans and bulk of cooking (WFH), calendars and school admin, a load of other things.
The best things that help are a shared Google calendar, a meal plan on the fridge, Muji 5-column calendar so everyone’s things are on view (and a column for bin day and random things!), splitting pickups drop offs, mutual understanding that it’s a struggle for us both, regular nights out with my friends who are all in the same boat, and a thoroughly decluttered house (I’m still working on convincing DP to streamline his fucking wardrobe so there’s less STUFF. No one needs 71 T-shirts and 60 pairs of socks! Less stuff, less washing, less putting away).