Every local authority will be different. Where I am you would currently be seen as adequately housed and so would not be very high up the list. You’re banded either gold, silver or bronze and at the moment you’d be bronze until the baby was around 2.
You’d have to get your mum to write a letter to the HA stating that she wants you to leave and giving a date you have to be out by. You’d then have to go into temporary accommodation, but you would then be banded as gold and you would probably get a place fairly quickly. This is how everyone I know in your situation has managed to get a housing association house.
A friend of mine did manage to move from a two bed to a three when she was only banded as silver (three kids sharing a bedroom), but she was lucky that no one else bid on that house, or she wouldn’t have got it.
As I said, every area is different. In some areas it will be easier to get a council house and in others much, much harder.