You can definitely hear a difference in the Queen’s Christmas speeches over the decades. In the 1957 Christmas speech video, the accent sounds a little archaic, but by 2015 the Queen has the same charming accent as Helen Mirren. It’s still RP, but a more modern form of it.
I should add that the context of the Christmas speeches has changed. The 1957 speech is set up very formally with the Queen in full formal regalia. By 1968, she was dressed in a day dress and by 1986, she was broadcasting from the stables and her accent has shifted as well.
The article also points out that the Windsor social circles have become less isolated than in decades past. Thanks to the late Princess of Wales, her children and grandchildren have much more contact outside royal residences than previous royal generations did.
Even so, it is difficult for the public to truly ascertain how the Queen speaks “colloquially”. By design, she has created a very formal persona and does not normally allow the public to see her speak except in formal speeches. Even when she is interviewed, her speech remains very formal, although this 2013 clip does show her relaxing just a bit. However, she still uses the impersonal one very frequently to describe her own daily duties.