Reporting to you live from midway through a re-wire and it's absolutely fine... so far. However with four kids.... is there any way you could offload a couple for a few days, even if you can't all go somewhere? Are they at school? If they're at home during the day I think that would be hard, maybe wait til summer so you can all go and do nice things outside the house while it's happening.
We've gone camping indoors, in that we're wearing limited wardrobes/eating takeaways etc, just so as not to have to get clean stuff out and get it dusty (though kettle and tea mugs in full operation of course) - but with four kids I don't know how easy that would be? Also that's a lot of beds. Is it possible to set up a room for everyone/most of you to sleep in? Then seal it - towels/plastic over doorframe during the day so you've got a dust-free escape at night. The sparks can do that one last and you could move into a finished one.
I think it depends on how they're accessing the wires, too. Ours, the socket wiring is going under floorboards, with minimalish digging out of walls (concrete/brick walls though, so even minimal digging is relatively maximal mess) for new sockets and chases. The sparks think the won't have to chase lights - but if they do that might get a bit rough.
Definitely box stuff up as much as you can, and invest in good dust sheets. We've put everything in each room in a big pile in the middle, securely sheeted. They have gone room by room so some rooms have always been usable, in some sense. When they're working in one room, keep the door closed if at all possible. You'll still get dust seepage but it won't be billowing clouds throughout the house.
At the end of each day I am doing a serious hoover - hands and knees over the whole house with the brush attachment on the Henry. Which is a pain but makes the house feel much nicer overnight.
The cats seem to be managing fine. I'm keeping them out of rooms that are full of dust but to be honest they're not keen to stick around and investigate drilling noises anyway.
I work from home so I'm here all day every day. I will update if the lighting phase turns into a horrorshow but for now I'd say it's doable.... depending on the child situation.