Derek Jacobi's (Cosmo Lang, Archbishop of Canterbury) presence has got to be a Shout-Out to I, Claudius, which is about another stuttering monarch who succeeded to the throne unexpectedly, and it might also allude to him being Brother Cadfael.
Myrtle Logue is played by Jennifer Ehle, who was Firth's love interest in the series that made him a heartthrob. Although this movie only gives her and Firth a single scene together, they make a big deal out of it.
And then there's David Bamber's blink-and-you'll-miss-it appearance as the amateur dramatics director who rejects Logue. David Bamber is probably best known for playing Cicero on Rome, yet another statesman with a speech impediment. He also appeared as creepy parson Mr. Collins in Pride and Prejudice with Firth and Ehle. And he additionally played Hitler in Valkyrie.
Dueling Movies: The filmcriticosphere tried to force it into this with The Social Network, both big-hitting, big-budget biopics trying to nab Oscars and pretty much the only two contenders for Best Picture. This went right into the actual Oscars ceremony: Social Network's synth version of "Hall of the Mountain King" was the music for the opening montage while the title speech was the backdrop of the Best Picture montage.
Fake Brit: The Australian Guy Pearce as Edward VIII. Although at the time, there was no separate Australian citizenship, Australia was merely a self governing part of the Empire and this is a subversion for Pearce, who was actually born in Britain (his mother is British) and moved to Australia when he was three.
Fake Nationality: Jennifer Ehle was born in North Carolina. (Yes, this means that we have an American playing an Australian, an Australian playing a Brit, and a Brit playing an American. The mind boggles...)
Could be a form of Fridge Brilliance. Churchill was, especially because of his self-righteous defense of British imperialism in 1930s and other personal and political flaws, considered a repulsive villain by many of his contemporaries.
Helena Bonham-Carter has personally stated that she plays the crazy people well. The last time she was a queen, she yelled "Off with their heads" every five minutes. But as the Duchess of York, she's an eloquent, articulate, caring and supportive mother and wife.
Reality Subtext: The film's writer was himself a stutterer, who took great inspiration from listening to King George's radio addresses, with his parents assuring him the King once stuttered worse than him.
Shout-Out - Just a bit of one, in John Boorman's Hope And Glory. The family is listening to George VI give a speech on the radio and the father says "He's sounding better, isn't he."