I think all those fanfics which make Azula out to be a soft aunt kind of miss who she is. I think if she’s healed and come to terms with how awful her childhood was, she would be totally committed to insuring her nieces and nephews got what was denied to her, Zuko, and her friends as children, but also be fiercely committed to ensuring that her nieces and nephews learned what they needed to know to survive in potentially dangerous world, regardless of what their parents thought about this. She’s more of an “actions” rather than words sort of character, so I think her focus with dealing with the children would be less on being super soft and nice and pleasant and more on finding things she can do to make them feel valued and appreciated. I also think she would be very willing to confront their parents if she disagreed with their parenting decisions.
In short, Azula would be the sort of aunt who would get a Red Ryder BB gun as a gift, regardless of the wishes of the relevant parents.