Lyndsay just ignore the 1950's twattery, the 'oh won't someone think of the children' nonsense & the Troll Hunters. Ignore, ignore, ignore.
I wouldn't even bother talking to her tomorrow if I'd come home to that after paying someone to clean for 5 hours. I'd have the wine, calm down a tad, then text her saying 'I've just walked in the door, clearly talking is going to be a waste of time. Don't bother coming back'. I'd call a locksmith & get the door locks changed & call an agency for a temp cleaner. Then go somewhere nice for the day & relax. You need it.
Our cleaner used to wash the sheets while she was at ours, then take them home to iron them (as she had a steam press thingy). It's lovely having sheets that have been ironed & it's no crime to send them out, but there are less expensive ways of getting the job done (just as nicely).
We used to eat meals from 'Cook' most nights if not eating out.
We had the cleaner three days a week, 4 hours a day, plus ironing
There's absolutely NOTHING wrong with hiring in/out as much help as you can. NOTHING. However, there are areas where costs can easily be cut now that life has changed for you. As you have already noted, shopping habits, food waste, that particular laundry service & cleaning hours are right up there. I expect there are other things too... But definitely NOT

All of that aside, I totally understand you indulging the children so much since the divorce, but this 'cook to order' thing has to stop. Your nanny knows your children well enough to cook meals they can all happily eat, X is for dinner, that's it. It's fine to do a few peas for one if they truely hate brocolli or whatever, but indulging them to the point of your nanny being a short order cook for them all is honestly doing them no favours. I'd also make sure that YOU tell the children that this is YOUR decision. Own it, for their own good.
As for leaving their stuff all over the place & their rooms in a state, that stops too. They are young enough, yet also old enough, to be told & to change their habits. You cannot send them out into the world expecting to be picked up after. This too needs to come from you. Leaving stuff 'wherever you put it down' is just what kids are like, but you can't let them get to the stage where they expect 'the help' to pick up after them. You need to own this too, but also ensure your nanny is on board with making sure it's done. It's not about 'cleaning' it's about manners & puttng the glass YOU used in the dishwasher, YOUR shoes in the boot room, YOUR toys away... Their flat mates & partners will thank you for it
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As for rota chores, I wouldn't bother (it's a mixed bag re that on MN, but I'm not all for kids have chores unless it's necessary & in your case it's not). I'd EXPECT them to do anything they were asked to do though, whether that's emptying the dishwasher or fetching something from upstairs etc
Sorry it reads like War & Peace. You just sound so stressed... It will all be ok 💐