This is so hard to deal with. May I suggest a few things -
Shift your focus away from your Mum to your Dad.
I appreciate this may sound the wrong way around but you clearly can't get through to your Mum without causing her and you distress, so think about your Dad. Put things in place to support him. Cleaners, food deliveries, and free time. Whatever helps.
So if he wants to go a play golf once a week, get someone in to sit with your Mum so he can leave her safe.
Make contact with your local Alzheimer's Society / Aged UK groups
Absolute mines of information and wonderful support - for your Dad. My Mum wouldn't join in anything, but my Dad would. So he started taking her to a music / singing group and they both really enjoyed it. Dad still goes as a volunteer supporter.
Make contact with local social services and request a carer's assessment for your Dad
And be there when they come to do the assessment, so they get a full picture. For example you can be talking to your Mum in one room and your Dad can talk more freely to the SW in another. Make sure the SW gets the full real picture of the situation. They are the one's who can arrange care and support.
Make sure your Dad is claiming everything he can - Aged UK website is very good source of information.
I'm afraid you've already said it - it's a crisis waiting to happen. And sadly that's how the natural pathway for people with dementia and their families - a slow car crash that you can see happening but are powerless to do anything about. The only thing you can do is support your Dad and be kind to yourself. The clinical situation with your Mum will change over time and then you will be able to get people into help her wash, change clothes etc etc.
Please don't make anyone any promises about care homes. Sometimes, in fact many times, admission is necessary. Distressing potentially yes, but even more so if you've promised someone you will never allow it to happen. Give yourself that freedom.