Not all schools are over-subscribed, and actually sometimes those that are have people drop out over the year after application, so those on the waiting list get in. Call the school and ask if they have a catchment and what that catchment is, and how likely you are to get in if outside it.
I'm really sorry you're in such a shitty situation. I would at least get a rental valuation on the house, if the catchment situation is bad news, because quite a few rental places are scruffy and you may find you can cover the mortgage. No harm in getting a couple of agents in to value, at least.
If you went on the waiting list at the preferred school, would your mum be willing to live at yours during the week, just for the short term? That way you could have childcare lined up until there was movement at the preferred school?
And if your child is under statutory school age you don't have to start them at a school but the school has to hold the place, and you're still entitled to the 30 hours free childcare at a nursery or preschool, too. So depending on when your child is born, you could keep to the present arrangements for a fair bit while waiting for a place to come up.