I think obesity and depression are intertwined in very complex ways - I know that, speaking for myself, depression renders me incapable of caring for myself, emotionally, psychologically or physically - basically when I don't like myself at all, why would I look after myself.
But then that leads to emotional eating and eating the wrong things - which in turn makes me hate myself even more (lying in bed, hating myself because of what I ate that day), which in turn leads to weight gain, which leads to more self hatred, which feeds straight back into being depressed again.
Whilst I understand what @Aquamarine1029 is trying to say, and in some ways she is not wrong, I think she is ignoring or underestimating the paralysing effects of depression and poor self image. So I would disagree with her statement that no pill can sort your life out. Antidepressants are not a magic wand, but what they can do is get you to a place where you are able to care for yourself and like yourself a bit more, and where you have a bit more motivation to make the effort.
When you are depressed, it can be an almost impossible challenge to get out of bed and get dressed. There is so little motivation or energy in your budget, that everything is a struggle - even the absolute basics. And as a mum with depression, I know that you often end up spending most of your energy/motivation budget on looking out of the children, leaving less than nothing for self-care.
It sounds as if this is where @ladyshittersparkles is, at the moment, so I would echo the advice to go back to the GP and ask for more help. I have just come out of a really dark spell of depression, and what has helped me has been a combination of ADs - Mirtazipine which helps with anxiety and poor sleep as well as depression, and Prozac - both in quite small doses.
Do you have people you can turn to in real life? Friends or family who might be able to help out on a practical level or with supporting you emotionally, or maybe a local voluntary programme who could send someone to give you some practical help?
I know what it feels like, to feel so low, and to feel as if you are drowning in the clutter and mess of life with small children - I have been there. If you need someone to talk to, please do message me, and I will help if I can.