I understand your reluctance to push for a hospital admission but here are some things to consider -
even with a really good social service dept and a maximum care package from them - your Mum will not get 24 hrs home care input. She wont get constant overnight care presence. (Not unless the rules have changed since Jan 2021, when I was told social services don't do over night cover)
High levels of night time care trigger admission for care homes. My Dad was 'OK' caring for my Mum during the day, but it was difficult to get her to bed and she would wake and wander, including getting out of the house. You can't be up and alert 24 hours a day.
It will take time for Social services to assess your Mum - again even with a good dept, although most can get some sort of care in place pretty quickly once the assessment is done.
A maximum care package will look something like this - 4 visits per day, of about 15-30 mins per visit. The carers would for example get you Mum up, washed, dressed, breakfasted. Lunchtime visit - they would make a sandwich or serve a microwave meal. Early evening - as for lunchtime. Evening sometime (can be early) get your Mum ready for bed, put her to bed and leave.
Get that call into social services today and tell them you're going home this weekend (give them a day and time) and your Mum is vulnerable. Tell them you've held the fort alone for 3 weeks and you are physically and mentally at breaking point and about to loose your job if you don't go back to work and you live a considerable distance away. Don't expect your GP to do much (sorry even the nice ones ) - home care is purely a social services responsibility and you will have to make that call yourself.
Then at the day and time you have given social services - you get in your car and you drive home. Probably will be one of the most difficult things you will ever do. But all the time you are there, social services can't / wont class your mum as vulnerable enough. You being there, keeps her safe enough in their eyes.
Many many times, the situation has to get worse before it gets better. It's like watching a car crash in slow motion - everyone knows it's going to happen by there is nothing you can do about it and it's shitty, absolutely shitty
You will come across the term "capacity" and you will come to hate the term. Capacity is a good concept - it legally endows individual the right to decide what they want to happen to them. So the individual can choose to have a flu jab for example. However it falls down when dealing with people who can express an opinion about their care, but actually don't have the full understanding of what they are deciding and / or have the ability to take the appropriate actions to care for themselves. For example someone who has dementia and forgets they've left the gas on when making themselves some soup and then drapes the tea towel over the hob.
So how does this effect you - say a social worker arrives at your Mum's and after assessing the situation says to your Mum, well Mrs Willow, we can get a carer into you for 4 times a day 7 days a week starting tomorrow. Your Mum says I don't need a carer, I've got Willow here. No you says I'm going home tomorrow. Mum says, well I'll be fine, I'll manage myself. I don't want / need a carer coming in. And that's that. Care offered and declined by someone with capacity. You know your Mum can't manage. The social worker also knows your Mum can't manage. But neither of you can force your Mum to accept care she doesn't want. This is where and why you have to step away. The person who everyone thinks needs care, has to be allowed to fail. It's the law. It's not the NHS fault, it's not social services fault, it's not your fault and it's also not your mum's fault.
I'm sorry for the essay. I'm sorry it sounds bleak. But the situation is bleak and you're in the middle, trying to do right by everyone and it's taking it's toll.
Please for the sake of your own health and well being - make that call to social services this morning.