Seems like there is a lot of different thought processes going on here, and not a lot of actual straightforward communication.
You say the caregiver is interested in dating you two. Have you discussed with her how that would go? Would she only date you together, or would you all have individual dates as well? Would there be a long term goal of her moving in with your family or would she be free to date others as well and potentially start her own family?
Then your mom is interested in the caregiver and thinks they are moving towards a relationship. Is the woman even aware of your mom's feelings about her? Is she in any way encouraging this? If she is aware, does she plan on telling your mom, using very clear/direct language, that she is NOT going to date her? Or is she interested in your mom, too? How would you feel if she wanted a relationship with all of you? Have YOU told your mom that you're interested in caregiver as well?
Honestly, I wouldn't move forward until everyone's on the same page. I probably wouldn't move forward at all, though, because this woman is responsible for your mother's care. What if you date, things go poorly, and your mom's care suffers because of it? This is way too much "mixing business with pleasure" for my tastes, but it's worked before and I'm sure it can work again. I'm just not that big of a risk taker. lol