You have to start the antidote treatment immediately. Cyanide poisons cytochrome C. That stops the eletron transport chain necessary to metabolize glucose all the way down to 32 molecules of ATP. The sodium thiosulfate will scrub the unreacted cyanide from your blood. I'm not certain whether the "antidote" can actually undo any enzymes already poisoned.
The short answer is that you die in 3 to 10 minutes. The number of antidote kits and qualified personnel to administer them is miniscule compared to the number of people who might need them.