What is the symbolic significance of the clock, the song sung by the prole woman and the nursery rhyme about the bells of St Clements?
The song that the woman sings symbolizes how the proles are liberated in spirit, in contrast to the Party members; even after a lifetime of drudgery, as Winston reflects, the woman is still singing. 'The birds sang, the proles sang, the Party did not sing.' This succinct quote underlines the sense that the proles are natural, human, instinctive, aligned to the birds in their love and freedom of song, while the Party members are stiff, unnatural, denying themselves and others the freedom and joy of self-expression.
The nursery rhyme symbolizes a lost world, the lost past, as mentioned in the answer above, but it also takes on an added, and ominous significance when the closing lines are finally recalled:
Here comes a candle to light you to bed
Here comes a chopper to chop off your head.
It is no accident that these lines are mentioned when Winston and Julia are arrested. This seemingly innocuous nursery rhyme has a sinister ending, just as Winston and Julia's stay with the supposedly gentle old man Charrington has a grim finale, when Charrington turns out to be a member of the Thought Police.
In 1984, Orwell uses these items to convey a deeper, hidden meaning.
Firstly, the clock symbolizes the difference between the reader's world and Winston's world. This is shown clearly in the first line of the story when the clock strikes "thirteen." Although we know that this is 13:00, the time in the twenty four-hour clock, many readers would not be familiar with this method of telling the time as clock faces only show the numbers one through twelve. The clock, therefore, signals a reminder to the reader that this world is very different to theirs. It is a world of totalitarian rule.
Secondly, the Prole woman's song and the nursery rhyme are both symbolic of the past, of the way of life before the Party came to power. For Winston, this was a time of personal freedom when people could live as they pleased and be with whomever they choose, without fear of violence.
To Winston, these symbols evoke nostalgia because he is desperate to return to this way of life. His dream is to create a world in which the Party no longer exists, a world in which songs and nursery rhymes flourish.
In my opinion, all of these symbolize the past. They symbolize the world that has been lost and that Winston wishes he could get back.
The nursery rhyme reminds Winston of his childhood. It reminds him of churches (perhaps to point out that there is no religion in the current society) and more generally of places that no longer exist. His trying to remember it is symbolic of his desire to remember things that the Party says never happened.
The song is a sentimental one -- talking about love and feelings. These are also things the Party tries to destroy.
So these are symbols of the ways in which Winston wants the world to change -- things the Party has destroyed that he wants back.
Sorry -- forgot about the clock -- I think it's just that it is an old-fashioned one. That's another thing that's like the old days.