You are not a bad mum, you are dealing with so much tough stuff right now. 
I had to call an ambulance once not long after dad died, because mum has a heart condition and she collapsed. Luckily it was Xmas Eve so they came quickly! They spent lots of time with her and it was considered that it was the strain that caused the collapse. That was with me doing all practical stuff - bar cleaning - mum has a cleaner once a week now because dad did most of it.
So, I can imagine that she feels she needs practical daily help, is it care or is it a home help type situation? Can her friends do any kind of rota, that’s how we managed when I wasn’t there.
Re the depression, the ambulance people made an urgent referral to her GP, who wanted to give her an antidepressant but alas, she didn’t agree. I think it’s a norm now to offer them in bereavement cases like this. I’m already on them and my GP upped the dose.
As for you, are you okay with how much you are doing? I do understand about not having time to grieve, I used to stay for days in a row, come back home - about 90 mins away - and cry for ages. Then I’d have to “reset” to go back within a couple of days.
Another friend who lost her dad in a similar set up to mine, told me very bluntly “if you lose the competent parent, it can be worse than losing them both in an accident or something”. Is that your situation? Mum never even paid bills. So there was a lot of stress apart from the bereavement.
I will be a cliche and say it gets better with time, but it also needed very clear boundaries and honesty with me and mum.
So I’m wondering, can your practical tasks on a Saturday be outsourced? Can a kindly friend or neighbor do things like laundry? That will relieve you a bit.
Has she done anything ordinary? The first day mum agreed to go to the tea shop for a bit of a cake, that was a good start.