Are you concerned that your 6-year old is going to ask where the dad is?
I am pretty simple I guess. I would just tell her “Your cousin… is going to have a baby” and then see where the conversation leads. At 6 my children knew about families that were simgle parent families. Children that just had a mom. We never got into details, because they are personal, so if they said “why don’t they have a dad?” I simply said “I am not sure” and then continue to see where the conversation leads. Often I find that at that age if I let them lead with their questions, instead of me trying to explain, that they get their answers with out too much. As I can easily talk too much and confuse them.
At 6 I would not get into anything about “she made a wrong choice” or anything like that. It is very personal and details do not need to be shared. You don’t want her walking up and saying “you made a wrong choice and are having a baby?” or anything like that. I would keep it much simplier. And there is a good chance she will be more interested in the baby, having a baby to see, then the idea behind how the baby got there. Do questions come up when she sees other babies? Most 6 year olds are just excited about a baby, not as excited about the parents.
Considering that I looked very young, even through I was 24 when I had my first child. People thought I was in highschool and I got many “looks” and strange questions. I could not wear my wedding ring because my fingers swelled. It was a very strange feeling, but those questioning looks never came from children; always adults.