Consider two spacecraft A, B in a line a distance L from each other. As per galactic law, the small drones they send back and forth to sync their hard drives for the day cannot exceed some speed m (ambient galactic-stellar frame of reference). They are traveling at speed v < m. How long does it take for one drone to make it from A (the rear ship) to B (the leading ship)?
$0 + m\, r = L + v\, r$
We solve this for r (the receiving time). Which gives
$r = \frac{L}{m - v}$
Now, when does the drone sent from B arrive back at A?
$(L + v\, r) - m\, (t - r) = v\, t$
Which we solve for t (the total round trip time) to get:
$t = r + \frac{L}{m + v} = \frac{2\, L}{m}\, \frac{1}{1-\frac{v^2}{m^2}} =t_0\, \frac{1}{1-\frac{v^2}{m^2}}$
Where t0 is the time it would take for the drone to make the trip when the ships are stationary (wrt the ambient galactic-stellar frame of reference)

Now consider two hydrogen atoms in a high speed gas or something. From the perspective of a scientist observing them, the electromagnetic radiation the two particles use to communicate also only travels so fast (in the lab frame), and can only possibly exchange information between the two atoms at a certain rate which is identical to the above expression. For complex objects like a stopwatch, the entire system would appear to slow because the speed information is traveling through the system at is bottle necked by this effect

So over and above relativistic effects, you have a time dilation factor that's approximately $\gamma^2$