As mentioned above, I think the answer is "Jehovah Is My Shepherd", used continuously since the late 20's.
Honorable mention goes to "Let Your Light Shine" which has been virtually unchanged since the 1944 red songbook. I think it was called "Fidelity" in the 1944 and 1950 books, but the lyrics & melody are nearly identical.
Of course, "continuous" is a bit relative, since Booze Rutherford cancelled all songs and singing at the meetings in 1938, because it took too much time away from the preaching, in his opinion. Knorr had his numerous faults, but at least he did re-institute music at the meetings in 1944.