I don't know if it originally comes from the military, but I do know it is a common saying in the military. When watchstanders assume the watch, they must be at their station 15 minutes prior to the scheduled relief time for a watch turnover (summary of happenings during the previous watch) so the offgoing watchstander should be able to give a turnover and be off watch at the time the oncoming watchstander is scheduled to assume the watch. If the oncoming watchstander is not there 15 minutes prior to the beginning of the watch and instead arrives at the scheduled time they are late. If they are there 15 minutes prior to the watch, thereby being early for the scheduled time, they are on time.

It depends on the deck (interior or exterior), but generally all enlisted members below Chief Petty Officer wind up scrubbing decks at some point. Depending on the ship or submarine type, there may be a division assigned for maintenance of the exterior ship's deck. On a submarine, it's known as Deck Division, and is made up of non-nuclear enlisted personnel. On a surface vessel, there is a specific division called First Division, responsible for shipboard exterior maintenance, including the decks.

Interior decks are usually maintained by offgoing watchstanders or the division responsible for the space. Most ships have a daily/watch cleanup, and a weekly shipboard cleaning schedule, and assignments are given out by the Senior Petty Officers in charge of each division.