To plot a position by dead reckoning the navigator needs an estimate of the leeway - the difference between the ships heading and the direction the ship is actually going.

As I understand it every turn of the sandglass (half an hour) the heading and speed will be recorded from the compass and log. It seems probably to me that they also recorded the estimated leeway during that last period.

The leeway will depend on the 'point of sail' - the heading relative to the wind - the wind strength, and the nature of the ship. A ship sailing close to the wind in a blow will make lots of leeway. A ship with the wind dead astern will make none. One way to estimate leeway, in daylight, is to look at the wake. And I expect they would estimate it based on their experience of the ship. How they recorded it I don't know?

In the song I assume that the call of 'larbord watch' is for someone to heave the log, signifying that another half hour has passed and they're nearer to the end of their four hour stint. Or maybe it's to change the sails, and any activity is better than just 'clinging to the weather shrouds'.