When police ask the mother of an abandoned baby to come forward for her own welfare, are they being truthful?

Saw a story in the news this evening and it got me thinking. Have seen a few of these over the years and the police always ask the mother to come forward so she can receive medical treatment etc. But surely the police would also want to open an investigation and prosecute?

I can't imagine people in this situation sacrificing their anonymity. Sad story all round.