The receivers are inexpensive ($5-$10 for the kind of accuracy that's useful here) and it's not hard to parse the NMEA strings and PPS they output into a spooky-accurate internal clock. It only takes a few connections and an antenna to integrate GPS into an MCU like an ESP (or an SBC like a Raspberry Pi or a whatever).
Like, really: The hardware is ridiculously easy.
The only difficult part is the code. But as we can see from this posting, the clock-driving bits are already written and are available for use.
Just graft in the GPS parts instead of the NTP parts, add your DST/location rules if you really must (hint: that part is harder than it sounds), and send it.
(And if the code still seems arduous, then remember: This is the kind of work that a reasonably-focused person who is armed with a decent bot can put together over a cup of coffee or two, even if they don't speak C. It may be popular here to poo-poo the bot here, but it's completely OK to get some help. Don't let pride get in the way of having fun, learning things, and building neat stuff.
The tailor doesn't lament the invention of the cotton gin.)
My current house, with low-E windows, aluminum siding, and a metal roof? It passionately hates everything about GPS.
But in more-typical (stick-framed, asphalt shingled, vinyl-sided, US-ish) houses? I haven't had any trouble with my very inexpensive u-blox (or perhaps clone) GPS board, a presumed-decent GPS antenna that we were throwing away at work, and dainty little [IIRC] u.FL to SMA adapter to connect the antenna with. (I put this all together just to play with making a GPS-backed, low-stratum NTP server -- which was a much more-rewarding process than it had any right to be.)
It was bizarrely good, in fact: While it certainly saw more birds and presumably had better accuracy when sitting in a window, I had real trouble getting it to cease to operate. It seemed to lock on well-enough to provide time and PPS until I put the antenna into a windowless closet.
That said: The antenna that came with this cheap receiver was trash -- at best, 1/10. It was hard to make it work even outdoors on a clear day. I eventually got sick of looking at that part and binned it.
That gives you another weekend project, and you can reuse your DIY NTP wall clock!
If I am interpreting their request correctly, they want a wall clock that knows where it is -- and also knows what localtime is in that position on the globe.
GPS (plus some hairy lookup tables) can accomplish that.
[0] - https://www.nist.gov/pml/time-and-frequency-division/time-di...
It has automatic timezone. The article also explains why it isn't as simple as it seems