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.
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.