This is really more of an extended comment, than an answer. But my feeling about that rhyme is that part of the reason why Orwell used it in the story, was as another example of Winston being "betrayed" by something he trusted. Basically one of the themes of the book is inescapable doom, and even the things that seemed good and seemed "on his side", O'Brien and Charrington for example, turn on him in the end.
The book clearly shows that Winston was fascinated by the vanished (or vanishing) past, symbolised, for example, by the paperweight. For him, the past represented a better place, or at least a different place (since he didn't actually know what it was like - see his attempt to find out from the drunk prole). At any rate, the past was a place that was free of the influence of the Party. It was a time before the Party came to be, and therefore (at least in Winston's mind), it was a purer, more innocent time.
And hence his fascination with trying to find out more words to the rhyme.
But in the section where the Party finally "arrests" Winston and Julia (for lack of a better word), Charrington quotes the end of the rhyme, namely, the "chop off your head" part to them. To quote the novel:
And then another quite different voice, a thin, cultivated voice which Winston had the impression of having heard before, struck in; 'And by the way, while we are on the subject, "Here comes a candle to light you to bed, here comes a chopper to chop off your head"!'
And of course, Charrington, in disguise as the harmless/kindly shop owner, had originally mentioned the rhyme to Winston.
How it goes on I don't remember, but I do
know it ended up, "Here comes a candle to light you to bed, Here comes a
chopper to chop off your head." It was a kind of a dance. They held out
their arms for you to pass under, and when they came to "Here comes a
chopper to chop off your head" they brought their arms down and caught you.
So, it was as if the Past of which he was so fond, and which he valued precisely because of its absence of the Party, has betrayed him as well.
No doubt Orwell had multiple reasons for using that rhyme - it's very old - but he made it fit well into the plot of the story.
Incidentally, this usage of this 18th century rhyme, in a mid-20th century novel, is probably its best known use in literature.