She may not need to be elected but there has been on going push to do away with the monarchy for her whole reign. She is still around as head of state only because she has managed to not say anything politically divisive and not stick her foot in her mouth for 70ish years.
Did the Queen have the power to intercede on his behalf? Did she, knowing that he had been instrumental in the war effort, refuse to do so? It's hard to imagine that the trial would not have come to her attention.
No, although the Pardon Power now exercised by US Presidents is indeed there because Kings had it, in Liz's case that power like any others which were once the monarch's powers to exercise as they saw fit are today exercised by "her" government which was chosen by Parliament, elected by the people.
By the time Turing was convicted that power is in effect in control of the Lord Chancellor, part of the British Government. Today although the title is the same, a Lord Chancellor would likely be an MP (ie elected, albeit not to do that specific job) but at the time Turing was convicted I assume it was a Peer (so, not elected) but chosen by the Prime Minister of the day (who is elected).
The most recent notable exercise of this power by the British government was to reduce the sentence for a murderer who (during day release) tackled some terrorist lunatic and thereby likely saved some people's lives.
The first list I found of her meeting US presidents: https://time.com/5333083/queen-elizabeth-trump-visit-preside...