Sorry, this is going to be long, but couldn't resist answering.
One ds (5, reception), one dd (3, afternoon nursery). Still have fulltime nanny.
General cleaning/tidying: cleaner 2x wk, M&F, does everything, inc. ironing. We try to maintain in between, have silly high standards.
Shopping: almost entirely me, big weekly delivery plus extras as necessary. Always coming home in office suit hung about with carrier bags.
Cooking: sort of both in ad hoc way, mostly picnicing except at weekends (nanny does all children's). Weekend cooking mostly me, but probably equal division on tidying up (can't do it together, can't be doing with each other's system).
Laundry: nanny does all children's, super-efficiently. Dh does own gym kit & stuff which can't be ruined. I waste most of a weekend day doing mine & rest of his.
Kids: I do M/W/F/Sat mornings, he does Tues/Thurs/Sun. All negotiable except not my Sunday morning. Rest of w/e sort of shared except it's the usual story of I look after them unless he has a special plan (or I ask). I always used to do all the homework side, I've just managed to persuade him to start getting involved. I do almost all of the nanny/cleaner/school admin things.
Gardening/DIY/car: he does everything. Don't have garden as such but 4 roof terraces & windowboxes & he does them very very beautifully (also involving children in jobs). I have to try to remind self of this when grousing about shopping/laundry/being default childcarer.
Other: I do holidays & travel, he does bills & whether we should be changing our mortgage. He's also much much better about keeping our social life going.
The whole thing is a horrible nonstop militaristic logistic, but I truly think that if either of us stopped working (and he does freelance on top of his FT office job) we would just go looking for other pressures until we'd reached the same state.