I wonder if Homestart could help in these circumstances?
www.home-start.org.uk/needsupport/need_support
Or like others say, you just have to cope. And you will, though it seems hopeless at the time. This thought used to petrify me when I lived alone up north with a newborn and knew no-one. I used to leave little boxes of food and drink at floor level around the house in case I was too ill to move or dropped dead.
Having panic boxes always ready is a good idea. ie One box that contains clean bedding, towels, wipes, clean babygros, Calpol, so they're all in one place to hand.
Keep some cash in the house, so if you are too ill to go out, if your neighbour will not help by getting some shopping in for you, you can always ring a taxi firm and they will bring some shopping for you. But honestly, neighbours should bea first call.
I had gastroenteritis one year with an 18month old. I could not get off the sofa for days I was so ill. But I crawled to the kitchen to get water and bananas for myself and my toddler was still breastfed then so she didn't go hungry. All her toys were everywhere, and we both just slept downstairs on the floor with cushions and blankets, I couldn't have made it up upstairs.
I made sure my phone was always charged just in case.
You get through it, even if you have older children, up about age 4 they will be fine sitting there watching telly all day, getting their own snacks out of a lower cupboard even. It might last a week, but you do get through it. You just go slowly through all the motions of changing nappies, getting everyone fed. I think Flu and gastroenteritis are pretty much the hardest ailments because of the debilitation, you literally can't move, so you crawl, you manage, you know?
Mothers are built for survival, our instinct to nurture our children kicks in, you may be feeling like hell, but your instinct will kick in and you will make sure they are safe, fed, occuppied regardless of how close to death's door you feel.