this is all about technicality, surely?
Technically, the ex can't stop him from taking his child to see his new partner. But if she says "no, you're not seeing him," and he doesn't challenge that and instead goes along with it, then practically she has done exactly that.
It's very easy to sit there and say that legally there's nothing she can do, but there are enough men going to court to fight for access to their children to know that there are women who do exactly that and do block contact. Where the dad is in the wrong is by not challenging her. He can't claim that she can do what she wants unless he actually challenges it.
Op, no-one has flamed you, but I think it's fairly clear that your dp has warped priorities. His child is perceived to be at risk to the extent social services have been involved, his ex refuses to allow him to see him, and yet he doesn't have the time or the money to do anything about it? Really?
So, if when your baby is born, your ex decided that he was going to take her and keep her with him, introduce her to a string of other abusive women and stop you from seeing her, and imagine you didn't have any money, would you just sit back and say "oh, I don't have the time or the money to pursue this" and cry about it?
There is not a man alive who could keep me from my child. I would go through hell and back for him, and if I thought he were being abused I would be on social services doorstep until they did something. And then I'd be at the court until they did something. And if I had to sell my soul to the devil himself then that's what I would do.
There are women out there who obstruct contact with their dc' fathers, and it's entirely possible that your dp's ex is one of those. However many of those fathers do whatever it takes at whatever cost to have access to their children. It sounds as if your dp is not one of those.