The London airport would cover your health problem, if it were caused by something that the airport was responsible for. For example, they would cover your care if you broke a bone falling on their stairs. They will not cover a health problem that you happened to have while you were there.
If you do a lot of international travel, you should switch to a supplemental plan that covers emergency care outside the United States. Standard Medigap Plans C, D, F, G, M, and N provide foreign travel emergency health care coverage when you travel outside the U.S.