Do date-specific hats (such as "'Tis The Season") respect a user's timezone? I'm guessing not since the only region specific info in the profile is the Location field
Well the Stack Exchange knows your IP.
They know when your awake.
They know when you've been bad or good.
So be good for...
Wait I went a little of topic there...
All of the time based events on the Stack Exchange are usually based on UTC Time.
Most Stack Exchange events are based on UTC/Zulu time, just for consistency and convenience.
For this event, there are safeguards in place to account for timezone changes and spans. Don't worry -- as long as you meet the criteria, your hat should be awarded to you.
As far as I've seen, all dates happen in UTC; for example, the rep cap is for one day from midnight to midnight UTC. So I'm guessing that the day specific hats would work that way as well.