A B737 flight crew reported auto pilot failed on short final during a CAT III Approach resulting in a temporary loss of control. Flight crew diverted and landed safely.

Narrative

While working the red eye flight to ZZZ the weather at the destination decreased gradually until CAT III conditions prevailed; necessitating the use of autoland. Conditions were low enough and while fuel was still above the required amount to conduct the approach my FO and I agreed that the prudent decision in the event of a missed approach was to divert. During the attempted approach the aircraft veered off course at approximately 50-70ft AGL and resulted in a rapidly destabilizing flight condition in which the aircraft was banked and heading towards edge of the runway and the grass. I have no doubt that if we had allowed the autoland to continue we would have been in an accident. We conducted an uneventful go-around thanks to excellent CRM and diverted to ZZZ1. In ZZZ1 we debriefed the situation before talking to Operations; Maintenance; Dispatch; and Crew Scheduling. We wrote the up maintenance report on the autoland. When it became apparent that we would be unable to quickly depart again to ZZZ my FO and I decided to both call in fatigued. We agreed that the combination of all-night-flying; failed autoland; and subsequent rapid divert; had significantly tired us to the point where we felt safety would be compromised if we were to depart again.