Allan tells Sydney he has less than 12 hours to live if he doesn't get the antidote from the blackmailer on time, but he says the exchange is to happen "at noon tomorrow." For it to be noon tomorrow, that means it's before midnight that he tells this to Sydney, which means it's more than 12 hours before the prearranged drop off for the relic and he'll be dead before he can deliver it even if he can find it.