Before all the kiddies swapped germs habitually every fall/winter and in turn spread it to their families who carried it to work, what season or time of year was the most common for the spread of illness? I've read that it is in summer because of all the heat, but I've also read that winter and fall are the prime disease season. In this alternate world, there isn't a social structure like there is in the real world, children are kept in an entirely different facility than the rest of humanity until they are old enough to contribute to society and everyone else is pretty much on the same schedule all the time but don't interact a ton with each other (think, in the same room, but not conversing or touching a lot). I realize, of course that children are not the main cause of disease in the real world, but they do help to spread it rather efficiently so I was wondering if eliminating that aspect of societal interaction would change the time of year that disease generally spreads? Also are certain sicknesses specific to or more common in one season than others?
It depends on the disease. Most temperate-climate airborne diseases tend to spread more in the winter. People aren't 100% sure why, but theories that seem plausible are that's the time of year that people are indoors in close quarters with others the most, and that dry weather tends to dry up any protective mucous in the nose.
Fecal-Oral diseases like Polio and Cholera may tend to be most active in the summer in temperate regions, and in the rainy seasons in tropical areas, perhaps due to the rain washing infected matter into water supplies.
Blood-borne tropical diseases OTOH tend to be a big problem in semi-tropical areas in the summer when conditions are best for their carriers (usually parasitic insects like mosquitoes). In the tropics I believe they are perennial.
A special pandemic would be special though. One of the theories of the spread of the Black Plauge was that the rats (and their fleas) carrying it essentially followed trade routes. That means it would have been more of a problem during periods people were traveling with goods for the rats to hide out in (presumably good sailing weather on the coasts, and market/harvest time inland). But of course people fleeing the disease front would have helped spread it too.
So if you want to get down to the seasonal level of detail on your disease behavior, you need to first work out its transmission vector, and its seasonal behavior, then that of its hosts, then work up from there.
Bird migrations (towards the warmth as the cold season approaches and away from the warmth as the summer heats arrive) would seem to be wonderful opportunities for viruses to travel quickly. By riding under the feathers of airborne water fowl, water-borne diseases could jump from pond to pond along the way. If people then drank from those ponds...
Holidays, weddings and other gatherings would also provide opportunities for local diseases to pass from guest to guest.