"God Save the Queen"
"God Save the Queen" (at the time "God Save the King") was first performed in public in 1745, although some evidence suggests it may have been first performed 1740. The tune itself is likely much older than that. The interesting thing, and one of the difficulties with this question, is that it has never been officially recognized as the national anthem of the United Kingdom, either by royal decree or by act of parliament. To this day, the United Kingdom does not have an official national anthem. It is, however, the official national anthem of Liechtenstein, although with different words - theirs is called "Oben am jungen Rhein" ("High Above the Young Rhine").
The oldest national anthem to be officially declared a national anthem is "La Marseillaise". Written during the French Revolution, it which was officially recognized in 1795. Then banned by Napoleon in 1799, and again by Louis XVIII after restoring the monarchy in 1815. Then reinstated after the 1830 Revolution. Then banned again by Napoleon III in 1851. Then reinstated again in 1879. Then banned by the Vichy government in 1940. Then reinstated 1946 as part of the post-war constitution. And, just to make sure, it was again declared the national anthem in the 1958 constitution.
WHO GOT IT RIGHT: JP Weigt, Kay van Nielen, Iron Saul, Marika Thiessen, Marc Quinlivan, Bob Milligan, Bahamut, Tom Slocket, and Peter Phelan.