It's a lot easier for either the mother or the father (depending on who wants it) to get the court to order a DNA test if everyone is living in the UK.
The OP's child is going to be living on the other side of the world from his or her father. When and how is this DNA test going to be done?
Do you think the OP, who has effectively chosen to leave the UK for good and for her child to have no relationship with his or her father, is going to organise a court ordered DNA test to establish the genetic relationship between her child who lives in Australia (or wherever) and the father who lives in the UK? For what purpose? She seems intent on undermining the relationship between them so getting British citizenship for her child is unlikely to be of much importance to her.
What about the father? How does it benefit him? Even if he can somehow get a court in the UK to order a DNA test and a court in Australia (or wherever) to track down the OP and force her to give a sample of her child's DNA to prove that he is the father (and it seems unlikely that he would be able to do this, tbh), how would it benefit him? Even if he is proven to be the child's father he might still not be able to get added to the child's foreign birth certificate (and they won't have a British birth certificate) and be granted parental responsibility. And even if he could somehow be granted parental responsibility for a child born and living on the other side of the world, it won't mean the OP is forced to return his child to the UK. All it will mean is that he might be required to pay maintenance. So it's difficult to see what benefit there would be for him in forcing the issue.
That leaves the child. If the child reaches adulthood, or late teens perhaps, and knows that their father is British and lives in the UK, and knows who their father is, they might be able to track him down and persuade him to do a DNA test to prove that he is their father so that they can have British citizenship. But the chances of them having completely lost contact by that point, or perhaps of the child never even knowing who their father is, seem quite high.
This child will be a British citizen from birth but will most likely never have the paperwork to prove it.