There aren't recordings of the princes in every day speech so that wasn't so important. But if you look up the few existing recordings of them they sounded pretty much as the actors did in the film.

Re the impact on the country of the new King's impediment: I watched a video recording from the Empire Exhibition in Glasgow in 1938 and the crowd got visibly upset when King George VI had difficulty speaking, and were also visibly delighted when he did a good job, so I have no problems with the import the screen writers placed on the speech at the end of the film. It was a fairly clever way of pointing to the fact that George and Elizabeth boosted morale in the country incredibly by refusing to flee to safety, and by following rationing etc like everyone else had to do. Their presence, and the speeches George made during that time, were hugely appreciated by the people of the UK during the war.

Also: regnal names. Yes, they do get to choose them, to an extent. There's been background discussion over here for ages about what Prince Charles will call himself when he becomes King. Charles hasn't been that auspicious a name. Agree with you re the nonsense line of Albert sounding too German.