We had this with my mother. It is possible to force residential care but it is a difficult way to go.
My mother had Alzheimer's, a full diagnosis, everything. They organised carers but my mother hated them, she would scream at them and try to hit them with her walking stick, so they were very reluctant to come. My sister and I had POA and had access to all the finances, a lot of the money was tied up in the house but there were enough funds.
Because my mother could say "no" she was deemed to have enough cognition for them not to put her in care. This was despite the fact that she could no longer go outside (she would get lost/be out at midnight, undressed etc.) and the fact she had no idea who we (her daughters) were, or where she was or what day it was....
Nor was she feeding herself properly, she also fell down the stairs and was lying there for hours until a carer happened to turn up and find her, but then she point blank refused to be seen by a doctor. After the stairs fall, she was afraid to go upstairs at all, so - as the bathroom was upstairs - she wasn't washing and had begun to "do her business" in random kitchen items, such as saucepans. It was dreadful....she even had social services visit but they wouldn't do anything just because she could still scream "no".
This was the point we decided we had to force the issue. It was awful, my sister was in tears. We asked for a meeting and it was attended by the doctor, social worker, Alzheimer's psychologist, ourselves and two carers. They were still reluctant to intervene until my sister said we'd had enough, we were at breaking point. I also pointed out that if they did nothing, and my mother fell down the stairs again and broke her neck, I would personally be holding them culpable. This seemed to do the trick and they hastily changed their minds.
The reason they are reluctant to force care is that yes, the person does have to be sectioned and relatives have to consent to this. An ambulance and psychiatric nurses turn up to take the individual away and when the screaming starts they have to section and force them into an ambulance. In my mother's case they pre-informed the police and two police officers were present to ensure everything was done properly and as gently as possible in the circumstances. It is very distressing for the individual involved.
The individual is taken to an elderly psychiatric ward for assessment and usually spends 6 weeks there while being assessed. We went to visit every day; the reality was that although my mother wanted to "go home" she had already forgotten the trauma of her being there. And when we showed her photographs of the family home she neither recognised them nor even remembered it.
The assessment decides what sort of care is required and it allows relatives to look at finances and sort everything out. There are long waits for local authority care but if the family can afford it, the individual can be discharged to a private nursing home....we had to sell the family home to release the money to pay for private care, but usually a home will take the individual if they know money from a sale is coming in due course.
For my mother it really was the best thing....the home we found was very loving and caring, and we knew she was safe and happier than when she'd struggled on her own. As time went by she remembered nothing at all anyway, and became content and smiley...and eventually incomprehensible. For those saying care homes are a bad thing, I'd say not necessarily. Our mother lived for another 7 years in that home, she certainly wouldn't have lived that long outside of it. And she had a very gentle, peaceful and fully-loved last few years of her life. Looking back I would do exactly the same, it was the right thing to do in her particular case.
Care homes often receive such bad press and people are reluctant to 'say how it really is' when dealing with the elderly and dementia, but I hope my experiences helps to inform others having to face the same decision.