Use arduino - what size - to fit inside clock case.
With RTC - to keep 15 second intervals accurate?
No need to have wifi or any connection to NTP server - clock does not need to know the absolute time. Unless it has to know when DST starts and ends.
If time needs adjusting for odd small adjustments we need a forwards and backward button?
What about DST? How to make it move forward/backward one hour? More buttons? how to automate that?
Use time alarms library.
Most likely mains powered? Or at least a wallwart