My brother & his partner separated a week ago as she had cheated on him! They have a 13month old baby girl and my brother is on the birth certificate. His partner moved out of my brother's home & back in with her mum leaving the baby with my brother. He runs around so the baby can see her mother and her family but the mother won't make any effort! What rights has my brother got with him having the baby for this long on his own?

He has equal rights as the mother does. Since he is on the birth certificate, he's the legal father. Absent any court action they both have joint legal custody.

What is his real question? Why is he concerned? What does he want?

Agree, same as the mother, he documents all of this with evidence of what he is doing, he files in court for custody of the baby.

By your use of the word "mum" I suspect that you might be in the UK, or some part of the British Commonwealth. Where are you?