Oxidation occurs within a day if the sulfite (not sulfide) level is depleted.
Oxidized wine isn't necessarily "spoiled", just different (and well, not in a good way in my opinion, but it's drinkable).
Corks let in plenty of oxygen. Artificial corks continuously let in oxygen, so those wines have a limited shelf life (1-3 years). Natural corks allow oxygen only for a limited time, which gives those wines a longer shelf life (10-20 years).
The ABV will keep most unwanted microbial activity to a minimum even without sulfite, but that's entirely unpredictable and depends on a combination of your sanitation practices and luck.
Bacteria in the presence of sorbate without sulfite can produce a geranium off-flavor.
You can just ride it out and it may be OK or you could open the bottles and add sulfite and cork again.