Diagnosis
The oscillation motor gear is likely jammed by an obstruction, or the motor has failed.
How to Fix It
Unplug the heater. Manually try to turn the heater head left or right. If it moves stiffly, look for a foreign object lodged in the neck joint. Remove it. Plug in and try the oscillation button.
If That Didn't Work
The plastic gear on the motor shaft has likely stripped. This requires disassembly of the head unit. You'll see a small DC motor with a white plastic gear. If the gear teeth are worn, the motor assembly needs replacement.
Heads Up
Never force the head to oscillate manually while the unit is plugged in or powered on. You can strip the gear instantly.