I'm trying to understand the full situation here. I'm assuming - possibly incorrectly - that your brother doesn't have a paid job, @Dottyraindrops , and is in receipt of Carer's Allowance?
You can't control what other people do, only your own actions.
If I understand correctly, you don't want to provide care for your mother (and that might mean that someone else will have to take her to appointments etc). You do want some extra help with childcare for yourself.
The only way might be for you to stop paying for holidays for your brother and spending the money on the extra childcare you need.
It then falls to your mother to organise care for herself, if she has capacity. That might mean that she moves into a care home, thus freeing your brother to get a paid job (if he's not already in any form of paid employment - as I'm already assuming he's with your mother nearly 24/7 ).
If she owns her house, then it's probable that it will have to be sold to pay for a care home, although it might be possible for carers to come to her - I'm not sure how the system works in the UK outside Scotland (if you're in the UK), but an income assessment will need to be made. There's a distinct possibility that your brother might need to find somewhere else to live if your mother finally agrees that a care home is the best option.
It's a horrible situation, but I don't see what else can be done.
The other thing to take into consideration is that your brother might not always be fit enough to provide care for your mother.
I recall phoning my husband's daughter to explain that I had a health issue and that if anything happened to me, that care would need to be put in place for her father. Her response was 'You're not thinking of leaving him, are you?'
When I got the all-clear and let her know - but reiterated that anything happening to me would necessitate care, her response was 'But you're all right, aren't you?'
It then dawned on me that the thought that she might have organise care had never crossed her mind. I think that this is quite common in families - possibly because the thought of organising this is overwhelming.
Your brother really needs the carer assessment that's already been mentioned, so that Social Work are at least aware of your mother's needs. That way, if anything ever happens the emergency number for Social Work can be called and they can at least have a record of your mother's medical problems.