You need to step back.
This is not your life that is being decided about here, nor is what's happened your fault/responsibility.
Your god-daughter needs space and support to make her own choices - she doesn't need people making choices for her about how her life is going to be organised. Her mum, her sister and you all need to let her decide what she wants.
Does she want to be a parent to this child? If she does, then it would be cruel to force/pressure/"arrange for" her to be separated from the child. Can you imagine how you would have felt if you had been kept away from your child and had to watch relatives act as parents, because those same relatives (and god-parent) decided that was best?
If she wants to be a parent, her mum needs to help her to be, by helping her to plan maternity leave, antenatal classes, parenting support... Having a child is a huge upheaval to anyone's life, and she's going to probably find it harder than most first time parents to work everything out, but professional support is available from the health service, charities, social work, etc.
You and your friend might see her as immature and having messed up her life, but a large part of that perceived immaturity is probably because her life has been so firmly decided by other people that she hasn't had a chance to mature yet. Encourage your friend to give her that chance. She also hasn't messed up her life or education. Let her choose what she wants to do next, be it trying to stay on at the grammar school, taking time out from education to focus on being a mum for a bit, studying somewhere else, or something completely different. Whatever it is, let it be her choice. Otherwise you're all risking permanently damaging your relationships with her when she finally gets an opportunity to discover that she is capable of making her own decisions about how she wants to live her life.