The time zone on your Apple Watch will always follow the time zone on its paired iPhone (providing the phone is within range of the watch from time to time).
When you go ashore the iPhone will probably roam onto the local cellular, so it might pick up local time at each port call. However, CHECK before you go ashore because you can also set the time zone manually on your iPhone and it might still be stuck somewhere else in the world. The other problem is that your iPhone's idea of local time might not be the time zone your ship is using for when you have to be back on board to leave port. If you are not sure, ask the crew for a time check before you go ashore.
How your iPhone behaves while on board will depend on your cruise ship while you are at sea. Some have on-board cellular which includes time signal, others don't include time or rely on WiFi (no cellular, no time signal).