Is your mum's caregiver privately hired by your family as a private nurse/caregiver, voluntarily caring for her because she knows her, or is she employed by an agency like a home care/sitting service?
It is incredibly unethical, if she is not just doing it out of the goodness of her heart and as a friend. This woman has the right to do what she wants in her off time but as soon as she realised the caregiver-client relationship with your mum was overstepping boundaries and pushing limits, she should have stepped back, reported it to her boss to be documented, requested to be moved, or whatever, if she is an employee of a company.
As far as you and your DH, pump the brakes on your hormones. How long have you known this woman? I realise it is awesomeeeee to have this hot-bi-babe damn near jumping for joy at the prospect of possibly being part of a triad. But seriously what do you really know about this woman? Somebody is confused, leading someone on, or reading signals wrong. I wonder what impression this caregiver has been giving your mum. Who is to say that this woman and your mum did not have something going on? This situation is all kinds of messed up. The caregiver is claiming that she told your mum it would never be a romantic relationship, but your mum is saying it is progressing towards a relationship. Those are on two opposite ends.
I would not touch this. Find another woman. Go on OkCupid, speed date, check out your friends, or whatever. Somebody is going to get hurt if this continues, and by the looks of it, your mum will be the one who gets hurt. She stands to not only lose someone she really cares about but a caregiver.