@lordfeather It sounds like maybe post partum depression, which is more common than people think
Sometimes a new parent fails to bond with baby.
https://www.webmd.com/parenting/baby/forming-a-bond-with-your-baby-why-it-isnt-always-immediate#:~:text=Studies%20have%20found%20that%20about,it%20should%20come%20with%20time.
Sometimes the birthing process doesn't go well, and some women suffer emotional trauma as a result
https://www.pregnancybirthbaby.org.au/birth-trauma-emotional#:~:text=After%20the%20birth%2C%20it's%20possible,birth%20may%20experience%20birth%20trauma.
We live in a society that says a mom is an absolute awful person if she isn't feeling joy over baby, doesn't love to be with him, or admits the delivery was a traumatic event. This makes is very hard to admit these things to anyone. It's quite possible this applies to her.
Whatever her issue, you can't pressure or threaten her into talking to a therapist. If you issue an ultimatum & force her to go to appointments, it'll be a waste of time because she needs to be ready & willing for it to work. Therefore, I encourage you to approach this from a place of compassion for her.
I understand you're exhausted. The truth is that even with both parents engaged, a new baby is exhausting. It does feel overwhelming at times. Our species used to live in close-knit 6 extended family were always around to help. My advice is to reach out to family that live in your area. Do you have a grandparent or aunt who can help? Also, don't hesitate on hiring someone: a sitter, nanny, Daycare.