I think your anger is understandable jingl and it's hard not to feel desperately sad about this tragedy and to feel so frustrated that someone couldn't have done something to stop it. I am trying to withold judgment until the facts are known, but as someone who has worked in both child protection and mental health I'm only too aware of the gaps in both and more specifically, between the two fields.
The poor mother seems to have been kept in hospital for longer than the norm, which I suspect must have been due to her mental health issues as there is no reason to think she was not physically fit for discharge. I am thinking that she was probably being assessed by her usual team and perhaps consideration was given to arranging a mental health act assessment. However perhaps the risk signs were not there to warrant this. Not every mother with schizophrenia would be a risk to their baby or have inpatient care, particularly if other family members are supportive as they seemed to be in this case. I believe she left immediately after the baby's father left after visiting, and I have read that he said she was 'sleep deprived but happy', so it doesn't appear that he had any major concerns at that point. Having already experienced psychosis she would have been at higher than normal risk of puerperal psychosis, and this can come on very rapidly, so who knows what was going through her mind to make her walk out in such a way.
I am surprised that a woman carrying a baby could walk past members of staff without being challenged, but again I will try to reserve judgment on this as I don't know the facts. I may not feel as angry as 'jingl' but I do feel immensely frustrated and wonder how the hell this could have happened. I accept that the staff wil feel dreadful, but that is no reason not to hold a full enquiry without, hopefully, pointing the finger at individuals.