I think this scenario is everyones worst nightmare and I suspect that some people may lose their house.
May I tell you about our experience? It is probably not exactly the same as you are envisioning as it happens 25 years ago after OH was involved in a road accident and got an acquired brain injury leaving him with MCI and unable to work. It is actually only in the last few years that he has deteriorated further and been diagnosed with dementia, but, nevertheless he was unable to work , we had young children (5 and 7 years old), a mortgage and I was not working. It was a nightmare.
What happened was that we drastically downsized - from our lovely semi to a small terraced which drastically reduced (although didnt eliminate) our mortgage and I made sure we got every benefit we were entitled to - OH got sickness benefit (now ESA) and DLA (now PIP) which entitled me to Carers Allowance and it helped that OH had been medically retired and so got a small pension (now £10,000) a year. There may be other things that you would be eligible for, but we wernt at that time. To suppliment out income I took on a low paid , short hours job at the childrens school. You are allowed to earn a certain amount and still receive Carers Allowance (currently £116), so I made sure I was within this and as it was at the childrens school I didnt have to organise child care. OHs parents helped us moved and were also available as childcare in emergencies - less so as they got older and both developed dementia, one after another (I was not the main carer for either of them). It was not easy living on that sort of income - we didnt have holidays, the childrens school uniforms were secondhand (as were most of my clothes), ebay was a godsend and I tried hard to give a semblance of normal life. When OHs parents died, we used the inheritance to finally pay off our mortgage, which helped a lot.
It was hard and looking back Im not sure how we survived, but we did and still are surviving. It would be better, though, not to have to give up work entirely - try and go for day care and/or sitters so that you can work at least part-time. Financial assessment will only be looking at his finances and the house will be disregarded while you are living in it.