From the horse's mouth:
"Alcoholic beverages can be purchased at Walt Disney World Resort by all Guests 21 years of age or older.
Any Guest who wishes to purchase an alcoholic beverage at any site in the state of Florida—including Walt Disney World Resort and Disney’s Vero Beach Resort—must present one of the following valid forms of identification:
- U.S. state driver’s license (includes U.S. territories*)
- U.S. state ID (includes U.S. territories*)
- U.S. military ID
- Original passport
- Copy of passport (either on paper or captured on a mobile device), along with a government-issued photo ID** that can verify the person and birth date shown on the copy
- NEXUS card issued by U.S. Customs and Border Patrol (Canadian Guests only)
Please note: All Guests visiting from outside of the United States are required to present both a valid passport as well as another form of government-issued photo ID from their country of origin.
For all questions concerning our identification requirements, please call (407) 939-6244. Guests under 18 years of age must have parent or guardian permission to call.
*U.S. territories include Puerto Rico, Guam, Northern Mariana Islands, U.S. Virgin Islands and American Samoa.
**Foreign driver’s licenses or ID cards are not acceptable by themselves."
The above is not exactly clear, in one case it says guests from outside the U.S. must present a valid passport [not a copy] plus other ID, but it also says Canadians can present a NEXUS card on its own. Last time I checked, Canada was considered "outside of the United States". Based on the bolded/underlined part, from the UK you'll need an original passport and a second piece of government-issued ID.
ETA: I haven't had the problem, I'm 57 with enough grey to be beyond doubt "of age".
