The Underwater GPS does not need an internet connection to work, so the positioning will work perfectly when offline.
We use OpenStreetMap in the map view. This map will only be able to show the tiles that have already been loaded (ie are cached in your browser) and not load new tiles when offline.
This means that to show map when offline you need to save the tiles manually to the cache before going out on a mission:
Go to the area that you will operate in: use the mouse to move around the map, zoom to a different zoom level and repeat for all relevant zoom levels to let the tiles be cached by the browser. (This needs to be done on the computer/browser that you are going to use on the mission).

(Yes we agree: It would be a great future addition to be able to select an area and download all tiles automatically for offline use)