If birth control is helping mood swings, the problem is likely hormonal, and Prozac doesn't address that issue. If she's been on the meds for as long as you say, it's unlikely to cause nausea now, unless she started the birth control recently. Personally, I wouldn't want a 15 year old to be taking birth control -- playing with hormones is probably not the healthiest thing to do for anyone, other means of birth control are safer for health. There are natural ways to try to balance hormones if that's indeed the problem. But given she's been on Prozac for two years, at such an awfully early age, earlier than the drug is approved for though it is legal to do so, it sounds like whatever the problem's root is it's unlikely to be hormonal because it started so early. But who knows? By the way, certain vitamins containing minerals can cause nausea -- has she started a new multivitamin?
I have never heard of taking birth control for mood swings. In any case, I would just call your pharmacist and ask about the interactions of those meds. Good luck.