It seems to me that it is claimed that the phrase (or a variation of it) was uttered in many different events, from Yalta all the way back to 1935, and to different people (Roosevelt, Churchill, Laval).
Do we have actual proof that the phrase was even uttered? If so, when and how did it take place?
The story is, in my opinion, almost certainly apocryphal. The first mentions I have seen of it from primary sources of any kind is US media after the Teheran Conference of 1943. Specifically Time: The Weekly Newsmagazine. They reported the phrasing as "The Pope. How many divisions has he?". I find it strange for a number of reasons. First and foremost, Stalin spoke little or no english. He did try to teach himself german and english in his youth but didn't get very far.
He undoubtedly learned some more during the war but by all indications it wasn't much and he never officially spoke in English, he spoke through his translators, at Teheran I belive it was Vladimir Nikolayevich Pavlov. Now, Pavlov did speak english, in fact Pavlov spoke a lot of languages. He was fluent in both english and german but also spoke french, italian and spanish pretty well. So, we know Stalin didn't say it in english and a professional translator and fluent speaker probably wouldn't have used a starnge phrasing like "how many divisions has he". In fact one of the reasons Pavlov was considered a good translator was his ability to convey nuances.
Another possibility is that Stalin said it in Russian and it was translated by either Churchills or Roosevelts translator but that has the same problem. I don't know who Chruchill brought to Teheran but Roosevelt brought John Bates, a native russian speaker. Bates using the same strange phrasing also sounds unlikely.
Anyway, after 1943 it took on a life of its own and appeared everywhere in a hundred different variations. Including the much better phrased version "How many divisions does he have?" which is what I would have expected either Bates or Pavlov to say had either of them translated it. However, this is all a year or two after the first mention and many of these mentions outright mention that it is almost certainly hearsay or legend.
It falls in the category of quotes that sound like something a person would have said, like Voltaire saying "I disagree with what you say but I will defend to the death your right to say it" (probably Evelyn Beatrice Hall) or Moltke the Elder calling the American Civil War "two armed mobs chasing each other through the country, from which nothing can be learned" (which he probably never said at all but sounds like he might have). Now, Voltaire, Moltke or Stalin might have said these things but there are no primary sources supporting it.
The evidence:
The evidence comes from U.S.A. Assistant Press Secretary Eban Ayers. quoting U.S.A. Fleet Admiral William D. Leahy during an executive lunch took place in 24 January 1946 in a White House executive lunchroom.^1
According to the excerpt, Joseph Stalin actually asked Winston Churchill "How many divisions did you say Pope has?" during the Potsdam Conference which was held in Potsdam/Germany to negotiate terms for the end of World War II between July 17 to August 2, 1945.
The context:
We don't know the exact details, but Eban Ayers gave a general idea in his excerpt:
According to W.D.Leahy (whom was a participant as Harry Truman's adviser at Potsdam during the conference), that was the funniest remark of the conference.
The incident occured during the discussion of Poland's situation. Stalin brought a proposal about Poles but Churchill objected that referring to Polish Catolics and said "Pope would be very displeased". Then Stalin turned to Churchill and asked that famous question "after taking a tug at his moustache": "How many divisions did you say Pope has?"
1 Cia.gov. 1946. An excerpt from the 24 January 1946 diary entry of Assistant Press Secretary Eban Ayers.. [online] Available at: https://www.cia.gov/readingroom/docs/1946-01-24.pdf [Accessed 14 January 2022].
