It has happened to us twice in 44 cruises, plus one close call ... not too bad, I guess. Both times were at embarkation.
Since I like to chat, settle back for all the details ...
First time. We booked a pre-cruise trip for 5 days in Iceland, and planned to fly to Heathrow and on to Southampton on embarkation day, knowing full well we were taking a chance. Two hours out from Iceland, the plane developed engine trouble and turned back! We called Princess from Iceland and were given contact information for their Heathrow representative when we finally arrived. Several hours later we arrived in Heathrow to find our ship had just sailed from Southampton! The Princess rep was helpful to the minimal extent of finding a black London taxi for us, but the taxi driver was a lifesaver. He drove us to Paddington station (yes, the bear place!) and double parked outside to take us in and show us the proper queue to buy a ticket to Falmouth, the ship's next stop. We had dinner and caught the midnight train and arrived exactly on time at 8:01 the next morning. We found a taxi to the port, and a Purser was waiting at the bottom of the gangway with a clipboard listing about a dozen people who had missed the ship! Our trip insurance paid the train fare and expenses, and Princess gave us credit for one day of cruising. The point is that we took a chance and lost -- BUT we took the chance knowing the next port was close and there was good infrastructure to get us there.
Second time: We were cruising from Santiago, and allowed lots of time by flying in the day before to be sure to make the ship. We flew to Dallas, on independent air not cruise/air. There we, along with about 100 other people bound for the cruise, were informed that our flight was delayed -- by 12 hours!! We gathered in the lobby and milled around for about an hour until we boarded buses and were taken (at airline expense) to a very nice resort hotel, given meal vouchers, and told where to get the bus back next morning. We flew to Santiago and arrived some time after the ship had sailed. Dubbing ourselves "The Lost Tribe of Princess" we began our routine -- wait around a while until somebody gave us some sort of information -- wait a while more for a bus. They put us up in Santiago the first night, meals provided, then next morning flew us down the coast to Puerto Montt. Another night in a hotel, meals provided, and next morning a Princess bus arrived to take us to the pier where a Purser's tent was set up to process all 100 of us aboard. It was all reasonably well handled, except for the inevitable delay and confusion, and worked out well in the end. All our expenses were covered (by the airline or by Princess? I don't know) and we had an adventure!
The third time involved heading for Fort Lauderdale to meet some friends and board the ship. Again we planned to arrive the night before. As our plane was about to land in Dallas (HMMM) it pulled up and the pilot announced 50 mph crosswinds so we were diverting to Austin. You can't get anyplace from Austin! Eventually we wound up back at Dallas as they were breaking out cots for the hundreds of passengers who had been diverted. My DW, being a TA, eventually pushed the gate agent away from her computer and started searching herself. Ultimately she got us on a midnight flight to Orlando. We arrived in Orlando about 4 AM and rented a car to drive to Fort Lauderdale. Getting there, we met the rest of our party at the motel. The small problem, however, was that our luggage was most likely still in Austin! I took our group of friends and we boarded the ship, while DW Renée kept the rental car to go out to the airport and search for our luggage. Several flights arrived from Dallas, but NOT our luggage. On the way to the airport, she had passed a Target store, so half an hour before sailaway she boarded the ship with four large shopping bags as our luggage! HAL was extremely good, offering us free laundry and even free formal wear rental. DWs shopping had been great and we managed very well. We had a day at sea, then a day at Half Moon Cay, so it was on the third day in Aruba that we were finally united with our luggage. HAL took care of everything and we just went off on a shore excursion and found the bags in our cabin when we came back! Our travel insurance paid for the extra clothes we had to buy.
So -- moral of the story -- when you travel, stuff happens. Try to plan ahead, but if it all falls apart then just go with the flow.
Now, I have to day that I do NOT recommend staggering out of Senor Frog's just as you see your ship start to move!! That may be a different story