She didn't have a 'home birth' in the normal sense though did she? Intentional home births are planned and have a midwife in attendance. Because the midwife is there, there is no need to go to hospital unless the midwife assesses there is a medical need.
If you have an accidental home birth in the UK and your midwife can't get there in the short space of time needed, then yes, it is normal to phone an ambulance so mother and baby can be checked over as soon as possible for problems which are not visible to the untrained eye but could be potentially life threatening.
I agree this story is very odd. I have had a look online and the name she gave to reporters is not the one she goes by (or ever seems to have gone by). We will never know because, of course, what happens in the future is private for the sake of the child. But it seems like the professionals did their job and there is definitely something not quite right happening here even if the baby is hers. I suspect this may well be a case of absconding from social services, but who knows? It's all odd.