First the use of the term "atomic": I believe it is just a branding terminology from watch manufacturers

https://www.twentytwoten.com/666/will-atomic-time-watch-sync-australia/

So, I believe the poster is trying to have Mathematica generate a modified IRIG H signal which will be locally transmitted (minimal broadcast meeting legal standards) somehow to the watch.

https://www.nist.gov/pml/time-and-frequency-division/radio-stations/wwv/wwv-and-wwvh-digital-time-code-and-broadcast

https://irig.org/

This signal will reset or synchronized "radio-controlled clocks" the preferred term by NIST:

https://www.nist.gov/pml/time-and-frequency-division/radio-stations/wwvb/help-wwvb-radio-controlled-clocks

In the US we have two of short-wave stations that broadcast the signals which can be used to synch radio-controlled clocks.

There is plenty of information available on the internet on the subject of radio-controlled clocks and the stations that broadcast the signals. One can start here:

https://www.rtl-sdr.com/wwv-and-wwvh-special-messages-to-broadcast/

Follow the links in the comments, watch the video, etc. The most informative was the following video

https://www.youtube.com/watch?v=6SHGAEhnsYk The youtuber provides many links. Most notable is

https://github.com/hzeller/txtempus

Now there are other solutions that can be found online none found using Wolfram Language.

Radio JJY Time Pulse Emulator

https://giammaiot.blogspot.com/2019/09/software-defined-radio-time-pulse.html

These sites as asides: https://www.youtube.com/watch?v=sdCLNtXDJb8 https://wwv.mcodes.org/

When or if time permits I may make a go of it but with no way to test I don't have access to a radio-controlled clock (watch). First function that comes to mind involves SquareWave[] or the like.

Wolfram Data

EntityClass["BroadcastStation", 
  "City" -> 
   Entity["City", {"FortCollins", "Colorado", 
     "UnitedStates"}]] // EntityList

WWV is not on the list. Yes it is short-wave, but.

I think the US does at least a decent job of providing accurate time at: https://www.time.gov.

That said, I needed a challenge this afternoon.

How about this (it seems to match https://www.time.gov):

First cell:

now = DateString[Now, "ISODateTime"];
Dynamic[now]

Second cell:

i = 1;
While [i < 10000000, i++; now = DateString[Now, "ISODateTime"]]

Though you will have to force the engine to stop.

What do you mean by "real-time time"?

Why do you need Mathematica to do this? On a Linux or Mac OS shell

cat </dev/tcp/time.nist.gov/13

Your computer is probably set to synchronize its clock with NIST so you can just do this from the shell

date

I feel there is something probably still missing from the description of the problem, but when I want something to show me the time, I use a cell phone or maybe my computer's clock, which set their times from one of the NIST servers. In Mathematica you can use DateString[] or DateList[]. I don't know what its source is, maybe the computer's clock.

Do you mean a Clock[] or an animated wall clock, or some other sort of clock, like a game clock? Why do you need two clocks: cannot the output of one clock be used in multiple places? Or is one a physical clock, controlled by Mathematica?