The confirmed oxygen toxicity cases I have looked at suggest it takes more than a few minutes. Here are some of the cases.
1. I talked to the buddy of a victim. They were diving the HydroAtlantic wreck off of Boca Raton--deck at 150 feet and sand at 170 feet. The hit happened after about 20 minutes. He thought his double tanks had air, but they turned out to have 36%. (PPO2 roughly 2.1)
2. The cave diver at Ginnie Springs started his dive on his AL80 stage bottle, which was clearly labeled oxygen. His friends had pointed that out, but he insisted that he had filled the tank himself, and it had air, not oxygen. He refused to test. It was pure oxygen. I don't know exactly how long he was at 100 feet before he toxed, but it was a while. (PPO2 roughly 4.0)
3. The WKPP diver was part of a group that planned to leave their 50% AL80s at 70 feet before going to the bottom at 200 feet. The diver took the wrong tank and was breathing 50% at 200 feet for a while before he toxed. (PPO2 roughly 3.5)
Calculated risk of pulmonary and central nervous system oxygen toxicity: a toxicity index derived from the power equation - PMC
The risk of oxygen toxicity has become a prominent issue due to the increasingly widespread administration of hyperbaric oxygen (HBO) therapy, as well as the expansion of diving techniques to include oxygen-enriched gas mixtures and technical ...

The like suggests that 15 mins at PPO2=2.0 is the limit for 1% risk of toxiing,. and goes up as the square of the time and the 6.8 [power of the PPO2. So PPO2-2.5 would suggest 7 mins as the limit.