I think its bloody good they are saying she needs to do a DNA because there are no witnesses to the birth.
Otherwise they could just be child trafficking. Never mind issues over which passport the baby is entitled to.
The idea that she could just rock up and deliberately give birth without any medical involvement is also one that merits Social Services getting involved when she gets back too.
Women who give birth unattended by accident are one thing but to do it deliberately is not clever. A homebirth with a midwife is a world apart from this shit. And potentially has safeguarding implications (again trafficking being on the list) for fucking off and leaving your child and for deliberately endangering the life of the other. When I was pregnant and asking for a ELCS, there was a long conversation about mental health and women who deliberately try and give birth alone without medical assistance (and basically - don't attempt it because its a massive red flag and we will get Social Services involved).
What the fuck did they think would happen if she got into trouble in labour too? It doesn't make sense. Not with leaving the daughter at home because they 'forgot to renew it'. Well I suppose 'forgetting to check the paperwork you need when giving birth abroad' is consistent with that level of fuckwittery. But yeah it all seems very 'off'.
Maybe THEY intended to sell the baby, and the sale fell through rather than them getting a baby on the blackmarket cheap.
I love how they think the Foreign Office should be doing more to get them back too. Well no, its a mess of your own making, they cant speed things up, you know what you need to do to prove you are the parents and the Foreign Office are NEVER financially responsible for getting you home in situations of fuckwittery (And this case proves why they don't pay otherwise every fuckwit and his dog would do stupid shit for months, spend a fortune and then scream HELP to get a free ride home!)