The letter, if exist, did not save Edith Cavell. She was executed by German soldiers on October 12th 1915.
However, Hugh S. Gibson, First Secretary of the U.S. legation at Brussels, did warn Germany consequence of the execution. It was prove later that Cavell was a British spy. Cavell then was use as an major propaganda tool by the British during WWI.
Uncle Sam is a personification of USA or more specifically the US government. The personification was use to recruit soldiers for WWI.
Uncle Sam is a national personification of the United States and sometimes more specifically of the American government, with the first usage of the term dating from the War of 1812. He is depicted as a stern elderly white man with white hair and a goatee beard, and dressed in clothing that recalls the design elements of flag of the United States—for example, typically a top hat with red and white stripes and white stars on a blue band, and red and white striped trousers. The first use of the term in literature is seen in an 1816 allegorical book, The Adventures of Uncle Sam in Search After His Lost Honor by Frederick Augustus Fidfaddy, Esq.
-- Wikipedia.