If mum is put onto a section III of the Mental Health Act then the NHS is responsible for any care home costs. However a section III does really take away all her and your power over her welfare and rights. It is given if the person is a danger to her/himself or to others.
My husband was moved onto a section III and spent the next nine months in a very poorly run assessment unit. I had to ask permission to take him off the ward and a walk in the gardens, to ask permission to take him to the hospital tea shop, was not allowed to take him off the hospital premises for the next six months. I felt that this greatly increased his illness and precipitated it into needing full time care.
Section III needs the agreement of his/her GP, the agreement of a consultant and in my case my agreement. Make sure you do know the implications of this section before agreeing. I had no idea that I was in effect, signing away all my rights to my husband's freedom, including where he would be allowed and if he would be allowed to go into a residential care home. It has to be reviewed after six months and can be extended for longer periods.
As far as I'm aware without actually checking up on DOLs legislation, Care Home costs would not be met by the NHS. DOLs involves the legal depriving of liberty or the giving of covert medication because it is deemed to be in the patient/residents best interest.
Whatever is decided about the two legal situations I would ask for an advocat to speak on your and your relative's behalf as either of these can be decided without your or your relative's consent.
Either will require a lot of careful thought as to what is best for your relative
xxTinaT