I haven't heard of any company that would commence payments before a baby was born.

I would definitely not be relying on this to happen, with out talking to whomever does your payroll - and understanding what their directive is.

I think you need to be looking at other ways of budgeting / finding the money.

Can your husband not get any more hours, either where is, or somewhere else?
Even casually somewhere - nightfill at a local supermarket?