Clear Skies Dragon Shout Refuses To Clear Winterhold

Not mentioned in any of the previous answers is that this is an identified bug, at least by the Unofficial Skyrim Special Edition Patch team. Identified as bug # 19800.

Clear Skies does not clear the weather in Winterhold despite all indications it should do so. This was due to a weather region setting that had snow storms set to 100%. This has been corrected to 99% snow storm and 1% cloudy, since the shout requires there to be a "pleasant" weather variety available. As a side effect of this, it will now be 1% possible in normal gameplay for Winterhold to have cloudy weather instead of a snow storm. (Bug #19800)

As this bug is fixed (and, for the record, I have the patch installed and it is definitely fixed), you can download the patch on Nexus by Arthmoor for Legendary Edition, and you can download the patch on BethesdaNet by Arthmoor for PC, XBox One and for PS4.

There are other things going on in Winterhold that could be affecting the weather. If you follow through the mage guild / college quest, you'll get an idea of what's troubling that region, and it's possible they'll elaborate further in DLC.

It's likely by design that you cannot use the shout there.