If you want a pure software solution get yourself an old atomic clock and https://github.com/jj1bdx/WWV play some tunes to set the time.

The repo you linked to is a WWV simulator, WWV broadcasts the time via _audio_ (double-sideband amplitude modulation) at various fixed HF frequencies. SOME clocks might be able to automatically receive and decode this signal, but not many. There is also a web version here: https://wwv.mcodes.org

Radio controlled ("atomic") clocks get their signal from WWVB, a long-wave station in Colorado. Its signal is just a carrier and data is encoded via pulse-width modulation and phase modulation. People have built local, low-powered WWVB transmitters to sync their watches and so forth in areas where WWVB is hard or impossible to receive. It's not a good idea to build one of these unless you REALLy know what you're doing because radio signals can travel farther than you expect, and the FCC takes a rather dim view of intentionally broadcasting your own signal (to any distance) without a license to do so.