Is it, like… not legal to give birth at home in England? This seems like a lot of press for someone who, ultimately, just had a home birth - possibly even an unintended birth outside of the hospital. Are all babies required to be observed by a physician or midwife within so many hours of birth?
Drawing on my social work training from ten years ago...
No, it's not illegal to have a home or otherwise unattended birth. You don't have to have anyone there or do anything you don't want to do, as long as you're not under any care plan for you or the baby that says you do have to be in a certain place and/or with certain people.
To me, the wording they use is less Baker Act (which I understand to be similar to being sectioned in the UK -- mental health treatment, normally against someone's will) and more their child is in a situation that has been deemed as potentially unsafe for the child.
That may be because previous children have been taken away, or risk factors were identified before and during pregnancy that meant the child would be under a care order (used to be the 'at risk' register but that changed since my training) which means that she did have to attend at hospital (or have a midwife/paramedics in attendance) to give birth, as far as biologically possible, and present the baby to certain care professionals for tests, monitoring, or removal.
Seems less likely, but it's also possible the mother had identified risk factors to her and the baby such as gestational diabetes or other pregnancy-related disorders (I'm not a medical professional so I'm sure they can fill in the gaps!) that means the police have concern for her and the baby due to medical, not emotional, social or physical reasons.
I think the wording would be different if there was an identified medical issue, with more focus on the certainty that checks need doing than the vague idea of something could be wrong as it could be with any otherwise healthy-seeming pregnancy.
She would be far, far from the first mother to go on the run with a child to try and get out of the grips of social services -- rightly or wrongly. However, I think her case got more attention than some because she didn't just disappear from home with her partner, but they had this mysterious breakdown that left them in a vulnerable position with a newborn, and that they haven't reached out since to even say 'we're fine'.