From a practical stance, you should contact Adoption UK as they are the experts and can advise you on the process.
But, I know a family where the mother was known to social services. She wasn't coping for a whole variety of reasons, but the children were at risk. Her father and his wife, took the children, at first to give the mom some respite, with a view to trying to get her reunited with the children.
It didn't work, and the children were eventually adopted by the grandparents after they had a residence order in place for over 5 years. This is the very abridged version as there was considerable heart ache on all sides, and lots of court visits to the Family Court, CAFCASS reports and social services intervention.
Social Services will only take a child at birth if there are some really exceptional circumstances, as mothers and newborn babies should stay together if at all possible.
It seems to me that there are 2 separate issues. One of you being concerned as a grandparent wanting to maintain a relationship with the child and also wanting to support your son. You may want to seek legal advice about the process where the father, whose consent is needed for adoption, is a minor.
In an ideal world, the mother would get support in a family environment, and be able to fall back on a support network. But most importantly is the welfare of the baby.
It may not be a bad idea to discuss the situation with a family lawyer as there are several issues to consider. You may also wish to post a thread in the legal section of Mumsnet with the query: 'How do I Adopt My Minor Son's Baby?' Or whatever title you choose. There are lots of Family Lawyers who know...
Hope it goes well for you and your family.