Are the school aware of your wife's situation? I know you won't want to go into lots of details but they really should be making an exception. Do you not have a social worker attached to your family or any sort of care plan for your child as a result? Your child should be accepted on that basis rather than your key worker status (although that would help too as you are of course unable to look after your child whilst working and your wife may feel unable to). I know it can be very difficult to move schools when your child has special needs but I would consider it if I was you as I don't like the sound of your school at all.
Both my Children have EHCPs in place. They are aware that things got very bad last time. My kids actually go to 2 different schools. I get the feeling one of the schools is just completely overwhelmed with the situation or just living in some fantasy land. Expecting us to home school and join Google meets with 2 Autistic children in the house trying to follow online learning plans from 2 different schools. I've basically given up at this point. Told the wife not to answer the phone tomorrow when they said they will be calling as I'll be at work and I don't want her stressed. My Children's education has now been ended and I suppose sadly with how bad things are they are just casualties of Covid that won't be in the numbers.
Education isn't only school. It does sound as though what the school.proposes isn't going to work though. I would try again to at least see if there could be a part time place.
We will keep trying, the worse part is they are brilliant with him that's why we are so devastated his education has taken away.