The Hell of Blazing Fires heats up the air in Ancient City enough to melt the falling snow (Subterranean Animism, stage 3). My only guess as two why snow falls at all is that the Ancient City was built near a former Cold Naraka.
The only formal explanation we have of the layout of the Underworld is Sanae in Wild and Horned Hermit, and she didn't go into detail about its biome.