I'm no expert, but I've not heard of any commonly available (read: "inexpensive enough for Fred to afford") 1930's-era navigation chronometers that were intended to be worn on a wrist or carried in a pocket, and there's a good reason not to. The only ones I'm vaguely familiar with would live in a padded box secured to the navigation table, to protect them from vibration, bumps, and temperature changes.
As I understand it, to take a sighting you clip a portable stop-watch to the sextant/octant, set to match the second hand on the chronometer, and click the stop at the moment you take your sight. Then you go back to the nav. table and add the stop watch reading to the chronometer reading to get the time of the sight. If you've got more than one stop watch, you can take more than one sighting before going back to your nav. table and working out the position.
The stop watches don't need to be as precise as the chronometer - they only need to accurately count the seconds for a minute or two. They aren't expected to keep precise time for hours and days - that's the job of the chronometer(s).
Hopefully Gary can clarify any mistakes I've made.