I used to work as a family law solicitor and dealt with a lot of these cases. If your friend were to see a solicitor (choose one who is a member of 'Resolution', ie a specialist and in favour of a conciliatory approach), then the first thing the solicitor would do would be to write to the mother, explaining that your friend would like to have Parental Responsibility and regular contact with the baby. Sometimes this can be enough to get channels open, without taking the next step of court proceedings.
However before he does that, I would urge you to get your friend to calm down for a minute and think about his own behaviour, as that comes through to me as the critical thing that is upsetting the mother here. Initially she allowed it; he stayed with them for a week after the birth and the contact has only stopped since then. From what you write, he and the grandmother were arguing over the 'right' way of feeding the baby, and he clearly doesn't like the grandmother or her partner. Think of the mum in the midst of this; a first-time mother, single parent. It's going to be tough anyway with a newborn baby with health issues; would you want people arguing and fighting around you? If she is forced to choose between her blood relations and the baby's father who is talking about his rights and telling her she's doing it all wrong, what would you expect her to choose? Your friend is clearly worrying and panicking because he cares for his baby, but he is going to have to deal with the problem of the influence of the relatives, even if he has court orders.
So my advice would be for him to breathe deeply, swallow his pride and his dislike of the grandmother and partner (however justified), apologise to the mum if needed in a nice letter, focussing on what he might be able to offer to do to support her. Even with on-demand breastfeeding, babies get hungry if the mum can't produce enough milk because she's tired or not eating enough. So offer to take the baby out in the pram so the mum can sleep, or bring/cook some nice food. Under no circumstances criticise the mum - she's probably doing the best she can. The more he can provide support, the less she will need to rely on her relations and the better the co-parenting relationship will end up being in the future. And they can be good, but it takes tolerance and understanding on both sides.
Good luck to your friend