Reset.it,
Do you mean a tire without tread, or a tire where the cord is showing? I can't speak specifically about Airbus, but in general tread isn't a deciding factor in tire replacement as it may be on an automobile. Most airline and large airplane manufacturers make use of retred or recap tires, and so long as the tire on the airplane isn't taken down below limitations for recapping, it can be operated. There's a lot of tire to lose before it becomes unsafe; you might be surprised just how thick they are.
Most operators won't ever let them go that far, but as a general rule the tire need not be replaced until cord or under-material is showing. Most operators will replace the tire or wheel assembly before that time as a preventative measure, however...it's far better to pull the wheel early, when one has a chance, than take a delay because one has no choice later, at a less convenient time.
Tires are consumables...there's really no need to let them go too far, especially if they're going to be recapped, anyway.