Bear in mind that this is an era in which access to past speeches, in recorded form, was hard: if it wasn't repeated on the radio, then unless you were obsessive enough to buy a record (which did exist, but were rare) then speeches only existed transiently on that week's newsreel or a live radio broadcast. We have access to the archives of Pathe now; people at the time didn't.

Here's George VI speaking in 1939. If you know he's got a stammer, then it's immediately obvious that it's a well controlled stammer. But it's hardly something you could imitate to amusing effect, because at face value it's just a slightly hesitant man with an accent of his class. By 1945 it's still obvious in hindsight, but quite how much of a topic for public discussion it was would be interesting to know.

There is a tendency in discussions of British social history in this period to confuse "everyone knew that..." with "everyone who knew Virginia Woolf or had slept with Henry Channon knew that...". It's hard to avoid, because if you turned around sharply in Russell Square you'd probably knock over half of the well known diarists of the era. You can see both sides of this in a fascinating (and extraordinarily well-written) undergraduate dissertation from 2010 [1]: on the one hand (p.52) , they quote Harold Nicholson (who knew Woolf intimately and lunched regularly with Channon) being quite critical of George VI's stammer, but on the other (p.53) they quote Mass Observation being far more charitable about both the King and the wider royal family's broadcasts. "you can't help but admire him".

The impression I have is that at the time of his death, there was general public affection, which transferred to his widow and his daughter. Mass Observation, from my memory of looking at some of it years ago, showed that even people sceptical about the institution expressed warmer views towards the individuals. That private affection, coupled with the deference of the era, would mean that if people knew about and were amused about his stammer they would have kept it to a very select audience.

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[1] https://scholarworks.wm.edu/cgi/viewcontent.cgi?article=1695&context=honorstheses