Judging from Skyrim, the dragons have a large effect on the province and it's people; burned down buildings, Helgen etc. What do you think the effects on the other provinces would be like? As bad as Skyrim? I don't know how widespread the Dragon Cult was, and the dragons seem to like their word walls, so perhaps they congregate around them? Are their dragons buried elsewhere?

What do you guys think?

Probably the same fate as Skyrim. With Kvatch in Cyrodiil having the same luck and getting burnt to the ground and not getting rebuilt for years, with its inhabitants content with living in tents. Also I think the dragon cult only controlled Skyrim, (This is quoted from UESP) Some time during or soon after the migrations to Tamriel, the dragon cult began to assert more influence and become much more malevolent. From their capital Bromjunaar in modern-day Hjaalmarch, they ruled Skyrim with an iron fist, effectively enslaving the populace. While what led to this change was lost to historians, it was likely caused by Alduin, the First-Born of Akatosh and leader of the dragons, who had chosen to forsake his role as World Eater in favor of conquering Mundus for himself. Man eventually rebelled in the late Merethic Era, leading to the ancient Dragon War.

This.
Also I believe dragons are in some way related to temperature.So a sudden rise then fall in temperature wouldn't surprise me, seeing as dragons returned to Skyrim, then someone killed a lot.