I actually think 'the Queen's English' was largely used in a different context from Amis. Following the two world wars (the first probably having the greatest impact) there was a significant shift in the social role of spoken English accent because of large numbers of working class men rising in the ranks in the army so that social status became more fluid. In the 1950s and 1960s the standard trend was for people of all classes to modify their accent to fit in with a spoken norm either called BBC English or Queen's English. Nobody actually tried to speak like the Queen (who clearly had a reason for sounding above everyone else) but they tried to speak like Richard Dimbleby or Cliff Mitchelmore on BBC magazine programmes. This was a strictly spoken concept, otherwise known as RP, or Received Pronunciation.
By the 1970s there was a backlash and by the 1980s it became fashionable to make a virtue out of a regional accent. The Queen's English went back to being considered 'posh'. The reason why this series of events is specifically associated with 'the Queen's English' is I suspect the much greater social role of spoken English broadcasting with the advent of television.
Amis, on the other hand, was interested purely in written English, for which the 'King's English' was the official norm - maybe dating back to the establishment of universal school education in the nineteenth century. I have never heard people talk of 'the Queen's English' in this context.