my brother-in-law has found himself in a similar position to your boyfriend. he was in a volatile relationship with a girl and they were on/off all the time. during one of the ‘on’ times she got pregnant. he suggested they abort as they obviously were not in a stable relationship but she refused. They broke up again twice during the pregnancy, but he kept going back as he didn't want to abandon his as yet unborn child.
Once the baby was born he stuck it out for another 6 months with her hurling the most horrific abuse (verbal and physical at him) on a daily basis. he eventually said enough was enough and broke up for good. She moved out (she had moved into his flat when pregnant) and has now found her own place but he pays maintenance and tries to get access to the baby weekly.
She now calls and texts him dozens of times a day hurling abuse, refuses to let him see the baby, uses the baby as a pawn e.g. he’s ill, you have to come over immediately. When he gets there, there is nothing wrong and she just stands and screams at him. She has damaged his car, called his employers and pretended he is harassing her (the list goes on and on). He is now a broken man and we don’t know what we can do to help him.
If I was to offer advice, I would suggest that he pays maintenance and doesn’t try to establish a relationship with the baby if his ex is anything like the ex girlfriend I have mentioned above you and your boyfriend could be in for a nightmare. While it is a shame for any child to grow up without both parents present in their lives, sometimes for the sake of each person (including the child who I shudder to think can hear his mother screaming and swearing at his father) it is better to cut ties.
Good luck!