OP, you're not in the US are you?
If that's the case we cannot help you... but if you're in the UK, I can certainly hand down some advice.
1. Proving "unfit" in the UK is the question, it's even more difficult do prove. Text messages will get thrown out before the parents can even blink.
2. She doesn't work? Will not be held against her. Moving around? Not held against her unless it's disrupting the child's stability and at 2 weeks old that's not going to be considered.
3. He hasn't established paternity, has he? Well, he's not going to have a leg to stand on before that happens.
4. Social services aren't stupid. I strongly doubt that they've ignored Dad's information - I suspect that there's no proof of wrongdoing.
That actually does basically match the US, so I'll summarise.
Unless it can be proven that Mom is unfit, the baby isn't going anywhere. Here's some things that WOULD come under the heading of "unfit".
- Mom neglecting the child. Neglect includes not feeding the child, leaving the child alone for more time than is acceptable, leaving the child in the same diaper all day and night, and things like that.
- Mom moving in or marrying a convicted child sex offender,
- Mom making meth in the kitchen while the child is in the house,
I think the point has been made, by everyone else, too.
This infant is not going to be taken from Mom without there being something exceptionally wrong.