Answer
The nickname "Toby" is associated with Edward William Barton, not Edmund Barton. Edward Barton, an Australian lawyer and politician, was given this nickname as a term of endearment by his father. As for Edmund Barton, the first Prime Minister of Australia, while he had several affectionate nicknames such as "Black Ned" and "Honest Edmund," there is no known record of him being referred to as "Toby."