@gral ,you are sadly right that Cathoic Church in 1962 was arleady infiltrated by masons,but maybe they manage to at least keep old latin rite.
I don't think they would keep the Tridentine Rite, because the theological justifications for the New Rite were already presented in Pius XII's Papacy -mainly in Mediator Dei, but also other encyclicals, like Musicae Sacrae. They will have to square that circle, and I think that the result will be a mix of Tridentine and New Rites, with the vernacular at the very least being used in some parts of the Mass(as was already done in Bavaria since the 1920s, and in parts of the former Austro-Hungarian Empire as early as 1906).
I'd expect other theological changes to go the same way, a mix of what was enacted after Vatican II and the previous norm - which of course would merely slow the process we see happening in the Catholic Church in OTL.
I think you can track certain left-wing infiltrations that ultimately led to disaster all the way back to
Rerum Novarum in 1891. Not to imply that Leo XIII was a crypto-leftie, of course, but his attempts to find a middle path while getting involved in such temporal socio-eoconomic matters would provide a precursor (and an excuse!) for errors... and deliberate warpings... down the line.
Sometimes, I think about a world in which Pius IX is succeeded by Cardinal Bilio in 1878. Of course, Bilio would only live until 1884, so most probably, Cardinal Pecci would still be elected, only six years later. But who knows? Maybe a few more years the hard line against all forms of modernism could still make a difference?
Let's dream. Pecci is still Pope from 1884-1903, but maybe with more of a focus on Thomism, and less of a focus on getting involved in socio-economic temporal affairs. That would be splendid.
Pius X was splendid, so nothing has to change there. He dies in 1914, and then... Serafini instead of della Chiesa? Would be a rather brief Papacy, only until 1918, but he was the most reactionary candidate, and therefore would be my choice for sure.
Then, perhaps, Merry del Val? Leader of the reactionary faction at the 1922 conclave in OTL. He'd live until 1930.
I have no real clue who would make an ideal Pope in a hypothetical 1930 conclave. Piazza? He'd live until 1957, which would allow for Cardinal Siri to become the Pope we never had in OTL from then until 1989. (Alternatively, August Hlond until his death in 1948, because having a Pope from Poland for the period 1930-1948 would be
something, for sure. And then just have Piazza elected in '48.)
Then have Ratzinger be elected early, and (in this more reactionary Church) stay on until 2022.
It's only a dream, but a very happy one. To see the Church fully dedicated to opposing modernism in all its forms, all throughout the 20th century; never once abating in its mission to keep the fires of faith and tradition alive. No leftism-infused social teachings, no Second Vatican Council, no Pope washing the feet of muslims, and no "inter-faith dialogue" that implies the ludicrous notion of equality between all religions.