Oh god OP, this is exactly where we are with MIL, and it's awful. I sympathise.
We found the Alzheimer's society helpline extremely helpful. It took several days before they called us back, but it was worth it.
Their advice is very good, and they can explain what you should be asking for and who you should be contacting.
We had spoken with MIL's GP who was quite helpful but they absolutely cannot tell us anything without MIL's consent.
We were very relieved to get a call from our local complex care and dementia team nurse out of the blue. We realised that the GP had obviously referred her to them without us knowing.
They seem very good and have rung MIL to offer help and an assessment which she of course has refused. They are used to this and are going to keep trying, and will keep us posted.
Don't worry too much about care homes just now. I would keep feeding information to the GP of any particular problems with behaviour and so on, and keep pushing for someone from adult care/social services to go in and carry out an assessment as your first step. Things do get a bit a better once they are in the 'system'.