Botulism as a disease is toxin-dose dependent. So it only makes sense that the more toxin the horse or any animal ingests and the more they absorb from the GI tract, the more rapid the progression of clinical signs. If it's a massive dose of toxin, horses can ingest it and be dead within 12 hours and never be seen to be abnormal by the owner. But that's extremely unusual. Most horses do progress over periods of 3 to 5 days, gradually becoming more weak and recumbent.
Most horses show some inability to eat normally or swallow normally.
Then once they get down, on occasion will get up and down for a few days and it depends upon continued absorption of toxins in the gut or how much they absorbed to begin with as to whether or not they will recover or go ahead and die. That is suspected to occur and not been proven to occur. But that is a plausible explanation. And then the toxins are produced in the dead animals. The horse has access to that animal or person or whatever - but usually the dead animal.