@destructive I am just coming back from almost exactly what you are going through.
I have never experienced such intense depression. As a parent, I felt intense shame and fear of being open with people about just how bad the places my mind was going to were.
What felt frightening was not realising or believing the intensity of what felt like a free fall spiral downwards would ease. Whist not linear, (bad days here and there), functionality and being able to be present with my children is returning.
I don't know how old your children are. For me it was possible to tell them in very child appropriate words that I was going through a difficult time and that I was going to be alright yet understood I may not seem myself.
I organised someone else to do the school run and sometimes cooking. I let go anything but basic cleaning, laundry. I started taking medication after speaking to my doctor. I I called crisis lines and also Sane, who I found really helpful. All conversations were out of earshot of my children. Sometimes I needed this support multiple times a day, a few of those crisis calls were profoundly helpful in getting through what felt impossible.I now am seeing a Psychologist.
I am trying to say, if you feel you can't talk to people around you, really reach out to every other form of help you can until you can feel able to put in longer term supports.
For three months I was disconnected, I made sure my children knew they were loved because that was about all I could do. Now, this past two weeks, I find myself laughing with the children, engaging once more. I really want you to know I understand and sit with you knowing how difficult this is.
It is not your fault at all, it certainly doesn't make you a bad parent.