The LA does a financial assessment taking into account all assets and income. House is included in the assets if not needed for either the person or their elderly partner to live in. LA then says how much they will contribute and how much the person needs to contribute. How you/the person actually raise the money is up to you.
If you can't raise the money without selling the house, and you don't have authority to sell the house or don't want to for some other reason, it is possible for the LA to put a charge on the house, so that they take the entire financial contribution once the house is sold.
If the person is entirely self-funding, no LA contribution at all, they may be able to get attendance allowance. But if the LA is making any contribution, then they can't get attendance allowance if they're in residential care.
So to answer the OP, if Mum goes into a care home, she would start by being self funding, paid for by a combination of her pension and attendance allowance, with the rest being paid for by sale of the house. Once the funds are down to a level of lasting only a few months, then you would seek a financial assessment and the LA would pick up the payment, (and Mum would lose the attendance allowance, and they'd also take most of her pension).
The only snag is if she's in a more expensive home and one which the LA wouldn't normally use, in which case they'd either want to move her to a cheaper home or ask you to make up the difference, hence the suggestion to check up on this first.