Recently back from the cruise that was the subject of this thread so I thought I'd add a few comments of what I actually experienced.
The ship had an local web site (it referred to it as an 'app' but it was actually just a web site) for accessing details about restaurants, menus, entertainment schedules, excursions, safety procedures etc. and also for raising questions or reporting problems. That was accessed via the ship's Wi-Fi network, for which there was no charge (but no internet connection, naturally). Turning off Wi-Fi would prevent access to that, so not really an option.
When entering coverage of different countries there was often a text message from O2Roaming announcing the fact and outlining the tariff e.g. "Welcome to GREECE. Calls to the UK and within our Europe Zone are charged at the same rate as your UK tariff". That didn't seem to happen in every region though: Albania was announced when just sailing past its coastline, and Montenegro when we moored there, but no announcement from the Europe Zone countries visited in between (Italy, Slovenia or Croatia) - no idea why not.
There were occasions when my phone reported that the available carrier was a satellite-based system ("GLOBAL NETWORKS" or "ship's mobile network") but as far as I could tell it never attempted to make any data connections without an explicit request. I just kept Wi-Fi ON most of the time, and mobile data OFF except when in port or ashore in a Europe Area country but it was rarely needed - Wi-Fi was available to customers at nearly all restaurants, cafes and bars anyway, and even as a free general (though patchy) service by some towns organisations.
Hope that might be of some use or interest to anyone doing something similar in future.
