Without ratting out your other neighbor (which I would really not be too concerned about that) the focus is the neighbor that is ill. I would just go outside one day when you see her outside and ask her if there is anything you can do for her and her family. You do not have to tell her anyone told you anything. Just by her wearing scarves and looking weak she will kno w you obviously know something is going on. When I was going thru treatment the best thing was when people ask me what they could do. You could get the family a grocery gift card, offer to take her to appointments or just ask her if she wants to talk to call or walk across the steet and you will be there for her. I am sure you know all this anyway having dealt with cancer recently with your family.
Good luck and make a move to her....
Jennifer