As dad you can ring your baby’s health visitor and arrange the help your baby’s mum so desperately needs. The Health visitor will arrange extra visits to support mum and go to the GP with mum if needs be. You can also go to GP and explain how mum is feeling.
Try talking to her again explaining that a lot of people feel like this after a baby and with the right help and support everything will be ok. Swap some of the maternity to paternity so can support her while she so desperately needs it, show her your in this together and will support her every step of the way.
Good luck to you both.