Most traffic between the "home office" and the ship is handled via satellite, either voice or e-mail. While the ships do have HF, it's not often used for business type communications.
Traffic between the ship, tugs and US Coast Guard will be via VHF. Traffic from the US Coast Guard command will be to the cutter on scene and then likely passed to the ship via VHF.
Transferring passengers between ships is dangerous and time consuming. Plus, you have to have an empty ship to put them on. Cruise lines don't keep empty/crewed and staffed ships sitting around. They would have to cancel other cruises to make that happen, and then it would take time to get the ship on scene. By the time they got there, the tug could pull them into port.
Either way, customers are upset.