sophiade2 you sound very sensible and kind and you are 100% right to support your sister.
I've read all your posts and as many of the others as I can. I've got a birth daughter and adopted son and understand a bit about relinquishment because I know a couple of people who adopted relinquished babies.
I think it is fair to say that social services will be most interested in the baby's needs, that there is never an automatic right to adopt a child so anyone who wishes to adopt this child would need to satisfy social services. If your sister knows or thinks her mum (your mum) to not be a fit parent she will need to tell social services.
The law does seem very confusing on relinquished babies. With children who are removed family are looked into first, with relinquished babies I am not sure it is quite so simple.
I think it is very important for your sister to really think this one through. I have little doubt that a healthy new baby will have very little problems in terms of being adopted and may well have a fabulous and happy life. Yes, they will need to come to terms with being relinquished and this can be very hard for some children.
Your sister, however, may deeply regret her decision when she is older. She will lose control of this in the sense that she may never see her child again. At the moment, this is presumably what she wants, but that may change.
It may be useful for her to work out why she wants the baby adopted out of the family. How genuinely feasible you or your mother's 'offers' to adopt are. (I am saying 'offers' because until you are assessed you would not know if you could offer it).
From the sound of it you or your mum could bring the baby up as your own and your sister would have a relationship with the baby as a 'niece' or 'sister'. Although you would need to be totally honest about the biological relationships for the sake of the child. Or potentially your mum could facilitate your sister bringing up the baby herself.
One of my relatives had a baby at 18, her parents helped a lot and all ended well.
Giving away a child is not a selfless kind act to some unknown family who will treasure it. Please do not encourage your sister to think about the people who may adopt her child and see this act as some sort of benevolence.
Your sister must 100% do what is right for her, IMHO. If she decides she does feel emotionally, mentally and physically ready to parent the child she must not think of giving her up because she may have a better life elsewhere. We just don't know how well her life may turn out, if her birth mum can cope, well, with help, this may well be best for the baby, IMHO.
However, if she honestly feels unable to parent the child, then she should be looking at other options. However, I would say this is a last resort.
Whether your mother is kind or not as a person generally, I do not know.
What she has said is harsh and wrong, putting pressure on her daughter. However, she may not just be thinking of herself, she may not just be thinking of her granddaughter. She may be thinking of her daughter and realizing what a monumental thing this is.
If your sister does choose adoption and it ends up being outside the family, please do encourage your sister to go for the most open form of adoption she can, at least letterbox contact with the adopters. Again this will not be something she can ultimately 'control' but if she is willing to write to the adopters and have limited contact this may make things easier in the long run.
Just be there for her. Encourage an openness if possible. You are doing your best, I am glad she has you. In terms of your mum, be kind to her too, try and stop it being Mum verses your sister. Both women are probably very scared but ultimately your sister has the right to choose to relinquish. 