My mom was served with eviction papers so she wrote to the clerk of the court and was denied because she failed to open an escrow account to show she had the money so she was denied an appeal so now what happens how long does she have before she has to get out?

Depends on the location. A few days to a few weeks. In many locations, once the eviction is ordered, the landlord can give as little as 3 days before the sherriff shows up and escorts the evicted off the property and has all of their possessions impounded.

The usual standard is 30 days.