slimboyfat : It still leaves the question why wait until 1962 though?
I believe that one of the primary motivations for the change in understanding of Romans 13:1 in 1929 (that the "higher powers" referred to Jehovah God and Jesus Christ) was to do away with the ambivalence during World War I whether it was necessary to obey the government in a time of war.
In the Watchtower of April 15, 1929 there is an article "God of Battle" with a section "The Christian and War" where it asks (para.21) :
But suppose the nation in which the Christian has his earthly domicile enacts a law compelling all persons within certain ages to perform military service, put on the uniform and go to war and use the instruments of destruction to kill. What shall he do ?
Of course the answer given is that the true Christian obeys the law of God, and six weeks later in the June 1 & 15 Watchtowers it was made clear that the "higher powers" did not refer to governments (and thus no compulsion to obey them). This meant that during the Second World War there was no confusion as to whether the governments had to be obeyed in time of war as there had previously been.
So while Fred Franz may have been eager to change the teaching once Rutherford died, there was a Korean War going on from 1950 - 1953 which meant the draft continued with the Selective Service Act being continually extended to 1953, 1955, 1959, 1963, 1967, 1971 and 1973. By 1962 the Eisenhower doctrine of "massive retaliation" demanded more machines and fewer foot soldiers. The Cuban Missile Crisis of October 1962 demonstrated beyond doubt that foot soldiers were obsolete on the international stage. The articles in the Watchtower in the following months (November 1 & 15, December 1) reverted the understanding of the "superior authorities" to refer to governments.
This is all just conjecture on my part, but I think it is reasonable that some or all of this was a consideration in the timing of the articles on Romans 13:1.