No, they don't adjust.
I state this because now dailies reset at 7:00 am (for my timezone). Before they were reseting at 6:00am. So, I lost a hour with the change, I have one hour less to make the daily writs before going to work. I'm at the EU server, in the GMT - currently BST.
Edited by redlink1979 on 4 April 2019 15:20
"Sweet Mother, sweet Mother, send your child unto me, for the sins of the unworthy must be baptized in blood and fear"
Sons of the Night Mother | VforVendetta | Grownups Gaming EU | English Elders [PS][EU] 2500 CP
Daggerfall's Mightiest | Eternal Champions | Legacy | Tamriel Melting Pot [PS][NA] 2300 CP
SweetTrolls | Spring Rose | Daggerfall Royal Legion | Tinnitus Delux [PC][EU] 2525 CP
Daily quests (writs, pledges, and other dailies):
0600 UTC for both NA and EU
Daily rewards:
0000 UTC for both NA and EU
Daily fence limit:
0300 UTC for both NA and EU
Weekly Trial:
0000 UTC Mon close, 1200 UTC Mon reopen on NA (12h after close)
1400 UTC Mon close, 0200 UTC Tue reopen on EU (12h after close)
Those are the times, regardless of time of year or location or phase of the moon or whatever. Ask Google to convert that time into whatever local daylight or standard time may be applicable to you.
Unfortunately, the start and stop times for events are keyed to 1000 US Eastern and are thus subject to the whims of US DST.