Ellen,
your health visitor is certainly there to ask about your physical and mental health, as well as the baby's.
She's not a housing officer, she's a health visitor.
Can you be clear and call her and say 'I need you to write a letter for me to help me to get rehoused' 'you need to write that It's over-crowded, which it certainly is, and is causing us stress, etc'?
Be very direct about what you need, she maybe stressed and overworked herself, so needs telling.
Step 2;
get to the childrens' centre and ask for a family support worker.
If they faff around, call and ask again, until you get one.
Having a dog is a big commitment, like having another baby, they need toilet training and have you considered how you'd walk a dog while having two babies on your own?
Maybe an alarm system is better.
Womens aid can advise you on getting help with alarms, after being a victim of DV.
Sadly, if your mums gives a kicking out date, you may end up in a hostel and bringing a newborn baby back to a hostel, which is unsuitable and depressing, and you'd risk losing any hope of housing all together if you didn't stay there 5 days a week.
Social services would likely be on your back too if you were in a hostel in your situation.
I would get your toddler a little bed for safety resons, and of course, child proof the bedroom you sleep in.
Most importantly you need a childrens' centre family support worker right now to advise you on all that's available for you and to be your first port of call for advise on everything.
Take care