I don't know the Watchville app but yes, it probably does synch with an atomic clock. Most (if not all) internet time services do.
But problems can occur in two ways:
If, for any of the reasons I mentioned, an app fails to synch. Then you're entirely reliant on the local timebase in the phone / pc / tablet / whatever to keep the clock right, and these local timebases can be surprisingly inaccurate. So, if it fails to synch for a day or two, you can be several seconds out.
The quality of the ntp or sntp implementation on the device itself. As an example, Windows uses sntp instead of ntp (a simpler version with less rigorous error checking) and Microsoft only specify the Win32 Windows Time Service as reliably synching within 1 - 2 seconds of GMT. For that, you'd do better with most ordinary quartz watches! I have no idea what the iPhone's implementation is like, whether Watchville has its own implementation, or whether it uses full ntp or sntp as Windows does, so can't comment on the quality of synch you might get.
The bottom line is that, without knowing exactly how the whole system has been specified, simply "synching to an atomic clock" doesn't necessarily mean atomic clock accuracy at any given time.
That's why I'd suggest getting a cheap digital alarm with "atomic" synching. That way, even with the cheap ones, you have a local timebase that's within 0.5 sec / day and 2 or 3 attempts at synching each day to a completely reliable radio source with virtually no latency - it doesn't take long for the signal to get to you when it's travelling at the speed of light direct from the nearest transmitter!