M.H.
If you went into labor, they would help you. But I'm not sure they would induce you unless it was a medical emergency (maybe your doctor would declare a medical emergency if you are a week late, I have no idea the circumstances of your pregnancy or your doctor's protocol).
You didn't mention your insurance status or if you are self-pay. Either way I would start by calling back the hospital and telling them upfront you won't be able to pay a lump sum within the week, and ask what your options are. I'm pretty positive you are not the first (and won't be the last) patient who is asking. Likely they will have options.
Good luck!