Okay, so some MASSIVE breaches of etiquette have been committed here by both you and your aunt's family, but at your own request, I'm not going to comment on those and instead address the situation with your 15-year-old cousin.

Assuming your parents are paying for the wedding and since it was your mom who did the uninviting, you just barely squeak by with a NTA, especially since you reached out to your cousin to let her know you feel terrible about what happened, she's still welcome to be a bridesmaid, but you understand if she can't/won't as her mother might not let her OR it would put her in a bad position of she did. (Note: if she went to any expense for being your bridesmaid you should reimburse her.)

Overall, though, this is a ridiculous, E S H situation; shame it erupted into a family feud.