Growth rate of the pathogen can vary depending on the species, but is also affected by variables like temperature, other stressors (like a secondary infection), and the immoncompetence of the host. Sometimes the infection can "explode" in as little as a day, sometimes weeks.
@RobK ... I don't think it is always that aggressive but yes, it can be. A week of treatment with no improvement and it getting worse says to me the meds you were using was not working so yes, it can kill especially a little fish I am sure different factors/circumstances can aggravate it at times like the stress of moving the fish to a new home, obviously water quality (which I don't believe it was your water but you don't know about the water he came from) I have found that if fish injure themselves in any of my tanks, it usually heals without any fungus but if at any stage I start suspecting or feeling it is taking too long then I treat Only time I ever had the start of fungus on an injury was on the tail of a baby ray (as I mentioned) but that was a deep injury and being a baby the immune system is not as strong and I think the stress also aggravated it Usually salt and Paraguard works perfectly For sensitive or scale less fish you start the Paraguard at half dose. You mentioned that he already had a small white patch so maybe he was hurt in his previous tank and the fungus had already set in there, with the stress of transporting, new environment etc.. it got the upper hand on the fish I think its just one of those things that happens sometimes but not every time.