Sorry, I took so long in answering! This was actually a cruise on Norwegian, but the cruise line was fantastic, as I'm sure Disney would be. They refunded our shore excursions for the days we missed. Welcomed us on easily when we caught up with the cruise in Malta (apparently this happens a lot and cruiselines have a way to embark you in each port) and had a plate of cookies and chocolates waiting for us in our room and a note welcoming us on board. There were over 50 of us with the same issue because of the same flight. We did not get refunded for the two days of the cruise we missed (we would have if we had
trip insurance), but I didn't expect to because it wasn't the cruise line's fault, it was American Airlines and their crappy planes and poor service. The American Airlines flight they cancelled was a flight from New York City to Barcelona, so it was largely made up of cruise passengers from all different lines. They stranded and ruined the vacation of about 300 cruisers. They could have brought in another plane and got us over there easily that evening or cancelled the flight earlier so we could find seats on other planes (we waited about 4 hours while they tried fixing the plane), but they didn't. They waited to cancel the flight till midnight and stranded us all till the next day. I have never forgiven American Airlines and have never not flown in the day before for a cruise since.