If the loss of her husband is very recent and raw do you think as the dust settles and she adjusts things might settle down? Maybe at the moment as she is in the early stages of grief, the thought of returning home is too much.
I think she would be better to stay in her own local area if not home with all her friends, social connections and known GP etc. Do you think with time and support she would come round to the idea of returning home maybe with some support.
I think the plan for her to stay for periods with different children is a bad idea for her and for her children. I think it would be avoidance and a sort of limbo for her.
Is her house suitable for her? Maybe you could ask the GP for an OT assessment to see how she manages cooking, dressing etc. You could identify where her needs were and what support she would need. This assessment might indicate if she needs carers in the home, sheltered housing or a residential home.
I think living between three of her children's houses would be a logistical nightmare with seeing her GP, OT assesments,prescriptions etc. There would be no continuity and things would be made even more difficult.
The idea of her staying with you is a definite no. It sounds like everyone is panicking including mil. Realistically you are in no position to care for your mil you already sound overstretched. And make no mistakes it will fall to you as the woman to do the caring- cooking,showering,hearing aids,GP appts etc etc it will all be piled on you.
I think the situation needs to be addressed calmly and slowly. Let the initial stress and upset of grief settle down, get her needs assessed and a full and frank discussion of where would be best for mil to live should follow. But you being her carer and having her to stay for periods is a bad plan all round.
Best of luck x