If an elderly person lives alone, has early dementia, falls is there any legal trouble for the family?

Can you provide more details, such as the age of your aunt, her current physical health and what cognitive symptoms she's having? If she falls in her own apartment and no one is her PoA or guardian, I'm not sure why you think that'd be anyone else's negligence? Which brings up the obvious: if you are that worried about her cognitively, physically and legally, why not try to convince her to move to Assisted Living where she'd have activities, medical attention, helpers and more social interaction? If she moves near your mom, does your mom understand that your aunt is expecting her to become her caretaker? Does your mom understand what this means? If your aunt has not yet or won't assign your mom PoA she should absolutely make it clear that she will not be her caregiver, as this lack of legal authority would make a difficult task needlessly more difficult and unpleasant. Just read the thousands of posts on this website. I hope your family can help your aunt make wise decisions for her future.

I’m interested in the answers to this question as well! Especially if undiagnosed and refuses assistance

At a minimum (if she is mobile), she should move to an independent living apartment within a retirement community so that someone could check on her daily and a transition to AL within the community would be easier when the time comes. Her cleaning and meals could be taken care of in an IL community.