NavList:

For celestial navigation 1 sec error in time will result in approximately 1/4nm error in longitude.  The minimum error that can be counted on in practice is 1nm, the optical horizon is only resolvable to about 1 arc-min.

The thing is with both mechanical and quartz watches, you can determine the error rate prior to departure and determine correct time within 1-2 seconds even 30,60,90d later.  For good mechanical and decent quartz timepieces, the error rates tend to be stable over the medium to long term.  If you know what the error rate is, what the error was on a certain date and how many days have elapsed since then it is a very easy process to determine the correct time right now within 1sec. In normal naval practice several timepieces and related documentation are maintained so that if one timepiece begins to wander erratically the consistent rates of the other timepieces will make that apparent and it can be disregarded.   Spending a lot of money on a super accurate timepiece is a waste for Celestial Navigation. 

If you want a super accurate timepiece because you want one that is a different story.