There are also terrestrial time stations whose signals might be easier to get than GPS in some situations. (Or not? I don’t know. I know they exist.)
http://genesisradio.com.au/VK2DX/time_signals.html
It would seem pretty simple to have one or two passive radio time signal receivers in the clock getting UTC and then have a literal knob on the front to set the local offset.
Is changing the time by hand twice a year really that big a deal?
And if so, shouldn’t DST just be abolished?
Building web servers into all the world’s timepieces does not seem like the right answer.