Mum of 4 kids here, and we also had my nehpew living with me. I can tell how we cope.
13kg washing machine, 2 10kg tumble driers.
Everybody in the house has 4-6 seasonally appropriate outfits. So now my daughter is in t shirt and shorts and has another set clean, a summer dress, but also has 2 leggins, a short and long sleeved top and a hoody. Two nightgowns. One towel each, one set of bedding each. Every child has the equivalent. if we have hand me downs that are too big they get washed and put away in a bin bag in the loft.
Every 4-6 months I go out and buy 10 pairs of the same sock in each sizes. I prefer all one colour or all the same odd colours so that any socks make a pair providing they are the correct size. All socks and underwear ( 5 pairs each) go into a communal undy box. only I am allowed into the box.
Every morning I have an alarm to remind me to wash the previous days clothes and one set of bedding and a towel. In my 13kg washer that is very often ( especially in the summer) just one wash. Its pegged out, or tumbled across two driers if the weather is bad. I then have another alarm set to remind me to bring it in, fold it and put it away. I only put the washing on the dining room table , nowhere esle. Then I have to clean it so my kids can sit down and eat.
Every night I set the following days clothes out when the kids are asleep. I get their socks and undies, lay them out and then collect all the dirty washing. You could collect it and wash it right then if you know you have an extra load the next day , and wait to peg out or dry in the morning.
The best thing I ever learned was to use the ikea kids storage for their clothes, the plastic box ones. Night wear in one, t shirts and tops in another, and pants/skirts in another. You can easily store toys AND clothes for 4-5 kids across 4-5 tall units. or use a huge kallax with the pull out boxes.
Declutter and get rid of ANYTHING that you don't needs, stream line it as much as possible. For tidying up I have a washing up box for each bedroom/child and it gets put on the stairs in the day time. Anything that I find, toys, clean clothes, books etc goes into the boxes and the kids take them upstairs and put them away at 3.30pm.
My youngest is 11 now but all of my kids could use the washing machine by 10 and all knew how to peg out and fold by the same age. My hayfever is bad today so 3 of my kids have kids pegged out, dried and folded 3 loads of washing as we are having a big sort out and bedding wash while the weather is nice.
I know its hard OP. Im bipolar. I have struggled and been overwhelmed at times too. But it really helps for me to set digital alarms and reminders on my phone, write lists with a top, medium and low priority that must be done that day and even if there is ten things on the list, as long as I do one of each priortiy then its a good day.