When a servant loses their soul, the also lose most of their capacity for rational thought along with the ability to use soul power. The little ability to think concludes that they must continue to survive and that they will need a soul to do so. Because of this, the zombie will attempt to kill humans to steal their souls, but since this does not work they will simply kill and kill until they are destroyed.
This may or may not be a more severe version of the same process that a servant undergoes if they attempt to use a Hyperstate beyond their capacity and lose control.
Lacking soul power, a zombie poses little threat to the life of a servant since they could only be killed by destroying their reaper. However, a zombie can still seriously inconvenience them, and if the servant is concerned about humans in the area they will need to keep it away from them.
A zombie retains their old power and if they had regeneration when their reaper died it will take a half hour to wear off as per normal. Between them, these two advantages coupled with a zombie's mindless ferocity can make it extremely difficult to bring down. However, their reckless behavior will quickly lead to injuries that will disable a zombie a short time after their regeneration wears off. If the zombie's head is destroyed, that will also kill it permanently, even if it would have still regenerated from normal injuries for some time.