Your mother is despicable. She’s furloughed and has every opportunity right now to teach her 16 year old to be a mum and give her some love and care. This is what your dsis actually needs. An adult to guide her. A parent. Your dsis is in her gcse year if I’m not mistaken. I understand your mother doesn’t want to be left holding the baby but this doesn’t sound like your dsis’s intention. My take is that this is your mother’s pride talking because she wasn’t consulted when you and your father were. She is a middle aged woman, who needs to get over her hurt and grow up.
As for your father, idk if he has the means to get a two bed. This would seem ideal. But tbh the one bed will be fine for now. Reading between the lines, I am wondering if he left the family home to your mother as you both lived with her and got himself what he could afford, which wasn’t much. He also needs to start being an equal parent to your 16 yo sister. I can see why your mother thinks she shouldn’t get the responsibility all over again.
Tbh the glaringly obvious situation would be for her to swap houses with your father. I know that won’t happen and perhaps I am wrong in my assertion that your mother is living in your family home. But can you see what I mean.
As for you, good on you for stepping up and being the big sister your dsis needs. You sound lovely. Unfortunately it is very unrealistic to expect your dsis to be able to study and look after her baby so please be aware this is going to be incredibly tough for her. Your mother knows this, which is why I’m saying how incredibly selfish she is being, and in what I think is a crucial year academically for your dsis. It’s hard to explain how all consuming a baby is until they arrive.
Someone upthread mentioned the boy’s mother. Have you been in contact with her at all? Maybe she would be willing to help with childcare. I know as a newborn your dsis won’t be able to be away from her baby for long. But it could be something to explore and a possible lifeline for her.
Is there anyone else, who could help? Aunts, uncles or sprightly grandparents? I know this isn’t a great time to be needing help. But I think you should try to help your sister find a support network. With vaccinations happening right now, the pandemic will hopefully soon be under control.