I'm so sorry to hear about your dad.
Honest answer to your question? No, I wouldn't live with my mum. This is something I've thought about a lot recently - my DM has early-moderate dementia, and my DF is now transitioning slowly from husband to carer. He's coping well, but I'm very conscious that the whole situation could change in a heartbeat. (He had his first proper fall last week, and although he's fine, he could very easily not have been.)
I've also watched my MIL care for my FIL with dementia at home until shortly before his death, and it's made me look hard at myself. Quite simply, I know that I wouldn't have the patience to care for mum myself. I already find it hard to deal with the endlessly repetitive conversations and questions, the permanent anxiety and need for reassurance and explanation, and that's just during regular visits and phone calls. And that's without (yet) any personal care needs. I just couldn't do it full time. I watched my MIL struggle similarly to care for her husband - she was often impatient and cross with him, because it's just so hard to stay calm and patient 24 hours a day.
Also, I quite frankly wouldn't be willing to change my lifestyle to accommodate caring, and perhaps this is selfish, but again I'm trying to be really honest. DH and I both work full time and have older teenagers. We're out of the house by half past seven, and we're back home at unpredictable times - anything from five until nine, depending on work and the kids' commitments after school. I spent 10 years as a SAHM, and while I loved that time, I feel like now is the time I get to live more for myself. I love my job, and the fact that I can work late when I need to. I love having the flexibility to decide on the day that I'm going to do something in the evening, because the kids can happily fend for themselves if DH or I aren't there. I love the fact that our lives now are busy and sometimes unpredictable. I love the fact that I can travel occasionally for work without needing to make a mountain of arrangements. If anything, I plan to dedicate myself more to work over the next decade, rather than less.
I'm just not prepared to change my life to the extent it would need to change in order to have DM live with me. I know that it will need to change a bit, and that more of my weekends will need to be spent supporting my parents as they start to struggle more. But living with them - even with full time carers - is just not a line I'm prepared to cross. When I was a teenager, I watched my DM make the same decision - not to have her DF live with us, after her mother died. I know she felt desperately guilty at the time, and I appreciate those feelings much more now than when I was 16 - but I also think it was the right decision.
OP, your feelings and your situation might be very different from mine, but at the very least I would urge you to only do this with your eyes wide open. Given the situation with your dad, you're probably not up for book recommendations right now, but if and when you are, I would recommend reading Keeper by Andrea Gillies. It's her account of moving her PIL in with her family, in a large house in a remote area of Scotland. Her MIL had mild Alzheimers at the start, which of course progressed. It's a very interesting read (she talks a lot about her research into dementia itself, as well as her own life), and a very sobering account of the massive impact of the decision on her family's life.
Very best wishes to you for the last days with your poor dad, and for the decisions that will come next.