It's a late 20th century phenomenon. The book was not translated into any European language until French in the late 18th century and wasn't completely translated into English until 1910 by Lionel Giles (as Sun Tzu on the Art of War). Machiavelli was certainly unaware of it. Also, the Chinese characters "兵法" are more literally "warfare" + "rules/method", so if anything it'd probably be more likely Machiavelli's work influenced the English title either directly or just by helping establish 'art of war' as an idiom in English.

Anyway, here's a telling statistic, courtesy of Google books on the frequency of "Sun-Tzu" in English publications. There's a small peak during World War II, where an increased interest in military strategy would seem self-explanatory (not least since China was at war). Interest seems to die down after the war and then there's a dramatic rise starting in 1964 and continuing to present day. This would coincide with the 1963 publication of Griffith's translation, which is also the first one titled simply The Art of War. Griffith's translation included more biography and commentary, but it doesn't seem too speculative to imagine that Griffith's status as an American general and war veteran also helped it garner more attention.