I've checked with
Disney Mom Jo and she's asked fellow UK citizens about their experiences.
Basically, fill in the
documentation that DCL provides online and then print it out.
Since you live in a country that participates in the U.S. Visa Waiver program and your visit is less than 90 days, fill out the I-94 green form usually provided by the airline. The I-94 white form is for non-U.S. citizens arriving with a visa.
Once you arrive the terminal, a DCL rep will check your documentation including your passport. It's a streamlined procedure that begins by entering the line for non-U.S. citizens.
When disembarking, all passengers collect their luggage first, then proceed to U.S. Customs. You'll need your Passport and Customs Declaration Form. This form is left in your stateroom on your last evening aboard the ship.
Immigration procedures sometimes change so please call Disney Cruise Line with specific questions prior to your cruise. More information can also be found at http://www.usembassy.org.