who pays court costs

I have no idea about this particular situation but what I do know is that even when costs are found against someone, they are never expected to pay the other person's costs in full. I wish you and your daughter good luck in sorting this out with minimum stress and expense.

I'm not sure on the cost's side either,surely if they are both unemployed they would have no way of paying fee's that built up.If the father is not interested in the baby at all can you not arrange with his mother to see the baby on her own,maybe at a park or such it would be such a shame that the baby and it's grandmother where denied a relationship because of the attitude of the father.

I suggest your daughter takes advice from a solicitor whose name is included on the Children Panel. You can identify experienced family lawyers on the internet.
My understanding is that this government has now removed legal aid from family law matters, meaning that everyone either pays or represents themselves.
The courts recognise that in most cases, children benefit from having a relationship with both their parents. There is a presumption of contact. It is very rare for a father to be denied contact and in circumstances like the one you describe, contact centres are likely to be the first point of call if the matter is considered by the family court. There are numerous contact centres, but not enough and i'm not sure how many will be able to keep going given the cuts.
Would the paternal grandmother be an appropriate person to supervise contact? If so, perhaps your daughter could sit down with the gran and talk through how this could be managed. I imagine your daughter would want to be with her baby during the initial meetings. I don't know where you are in the country, but there is an excellent service in manchester, called Pro Contact. Good luck, it's so stressful isn't it

the fathers mother is part of the problem, when my daughter ended the relationship, his mother demanded that my daughter pack clothes for the baby as she was going to have him or the weekend!! did not as but demanded, my daughter came to stay with me because she was afraid.
his mother said she would do everything she could to take her sons child, my daughter even had to leave her home and move in with me as they harrassed her. now it is his mother who is pushing him to get the baby he is not really interested. when they separated the baby was only 10 weeks old, now he is 22 months old and in all that time the babys dad nor his family have asked anything about baby if he well etc..in fact the fathers dad ignore when see them in street too. it is so frustrating.