Enlighten us about these supposed unusual circumstances.
There is only one unusual circumstance in John XXII case was whether he should be considered John XXIII or XIV because of the confusion of the existence of an Anti Pope who called himself John XXIII back in the middle ages. And Roncalli as Pope resolved the problem by saying he was XXIII.
The only unusual circumstance in Paul VI's election was the fact that he had been considered the leading candidate to succeed Pius XII because of his close association with him. Alas he was still only an Archbishop at the time of Pius death and the Cardinals decided to elect another Cardinal. He was close to John XXIII too. So when John XXIII died, he automatically became the leading candidate to replace him. He did. One of the few times when the leading candidate going in actually came out a Pope.
All kinds of moronic nonsense was concocted by the schematics of the SSPX stripe claiming that in both cases, the Cardinals had actually elected someone else. Of course they could never explain why the majority who lived after those two elections never said a word.
Amazing that you can come up with these silly, meaningless innuendos that have no basis in fact or reality.
Scortesco's assertions were published in the French periodical Introibo, No. 61, 1988. The Introibo article was based on letters Scortesco had written in 1976. These 1976 letters were in turn based on information Scortesco had received from his cousin, Prince Steno Borghese, a prominent member of the Vatican's Noble Guard, and “President” of the 1963 Conclave. Scortesco asserted that the threats were delivered into the 1958 and 1963 conclave by the highest masonic lodge, the B'nai B'rith.
“In the case of John XXIII (1958) and of Paul VI (1963), there were communications with the outside. It was thus known that there were several ballots in the first conclave  which resulted in the election of Cardinal Tedeschini and in the second , Cardinal Siri.” (Excerpt from the French Newsletter, Introibo, No. 61, July-August-September, 1988, Association Noel Pinot, Angers, France, p. 3.)
Scortesco was found burned alive in his bed shortly after the publication of this letter.
What is unambiguously known is thatVatican Radiodid conclude, on the basis of the five minutes of white smoke on October 26, 1958, that a Pope had been elected on the third ballot and announced it as such, telling listeners, "The smoke is white...There is absolutely no doubt. A Pope has been elected."
“ In 1954 Count Della Torre, editor of the Vatican newspaper L'Osservatore Romano, warned [Pope] Pius XII of [Cardinal Angelo] Roncalli's Communist sympathies. Other members of the ‘Black Nobility’ expressed similar concerns.
“Nor did Roncalli [later known as "Pope John XXIII"] escape the attention of the FBI and CIA. The agencies began to accumulate thick files on him and the questionable activities of other ‘progressives’ within the Vatican, including Monsignor Giovanni Battista Montini (the future Paul VI).
“Pius XII had appointed Cardinal Giuseppe Siri as his desired successor. Siri was rabidly anti-Communist, an intransigent traditionalist in matters of church doctrine, and a skilled bureaucrat. . . .
“In 1958 [on October 26], when the cardinals were locked away in the Sistine Chapel to elect a new pope, mysterious events began to unfold. On the third ballot, Siri, according to FBI sources, obtained the necessary votes and was elected as Pope Gregory XVII. White smoke poured from the chimney of the chapel to inform the faithful that a new pope had been chosen. The news was announced with joy at 6 P.M. on Vatican radio. The announcer said, "The smoke is white. . . . There is absolutely no doubt. A pope has been elected." . . .
“But the new pope failed to appear. Question began to arise whether the smoke was white or gray. To quell such doubts, Monsignor Santaro, secretary of the Conclave of Cardinals, informed the press that the smoke, indeed, had been white and that a new pope had been elected. The waiting continued. By evening Vatican radio announced that the results remained uncertain. On October 27, 1958, the Houston Post headlined: ‘Cardinals Fail to elect pope in 4 Ballots: Mix-Up in Smoke Signals Cause False Reports.’
But the reports had been valid. On the fourth ballot, according to FBI sources, Siri again obtained the necessary votes and was elected supreme pontiff. But the French cardinals annulled the results, claiming that the election would cause widespread riots and the assassination of several prominent bishops behind the Iron Curtain.
“The cardinals opted to elect Cardinal Frederico Tedischini as a "transitional pope," but Tedischini was too ill to accept the position.
“Finally, on the third day of balloting, Roncalli received the necessary support to become Pope John XXIII. . . .” (Paul L. Williams, The Vatican Exposed,Amherst, NY: Prometheus Books, 2003, pp. 90-92.)
The footnotes that Williams' references are:  Department of State confidential biography, "John XXIII," issue date: no date, declassified: February 15, 1974; see also Avro Manhattan, Murder in the Vatican, p. 31.  John Cooney, The American Pope, p. 259.  Department of State secret dispatch, "John XXIII," issue date: November 20, 1958, declassified: November 11, 1974.  The announcer's words appeared in the London Tablet, November 1, 1958, p. 387.  Houston Post, October 27, 1958, pp. 1 and 7.  Department of State secret file, "Cardinal Siri," issue date: April 10, 1961, declassified: February 28, 1994.
If you don't want to believe state department and other US intelligence agencies documents than I can't help you. If you want to keep your head in the sand and believe that no outside anti-Christian/Catholic parties or groups have tried to infiltrate and bring down the Church from within, again I can't help you.
@David: If you think that these cardinals, who are men, can't do something against Gods will than can I get a free pass to heaven too by being ordained one? As if I never did anything against His will I'd never sin and then that would de facto put me into heaven. Please realise they are MEN and CAN and DO, do things that are not in accord with Gods will or the Holy Ghost, just as we can make choices with or against His will. Just because Jesus said the Holy Ghost will always guide the Church does not mean men always elect the man who the Holy Ghost had chosen or we would have no such thing as free will.