Mom has advanced dementia and needs 24 hr. care, and no one has POA, what can we do?

So sorry that your mom is suffering from this awful illness. I am not an expert by any means but always trying to help out by sharing things that I've learned along the way in dealing with the need for elder care for both of my grandparents who suffered/suffer from moderate to severe Alzheimer's. I've read about something called an Elder Guardian that may be the answer here whereby the courts can appoint someone as a Guardian to make decisions and such. Try finding out more about Elder Guardianship for your particular state as the procedures do vary from state to state. If you have access to a lawyer, definitely share the details of your mom's condition with them along with the fact that there is no POA in place and that someone needs to be able to make vital decisions on your mom's behalf going forward. Give that a try; Elder Guardianship.
Hope that helps.