It makes sense to me that you would say that the song the woman sings as she hangs up the laundry has significance. I think that the overall message of her song is similar to the message of the book, even if Orwell says that the song was made up by a machine.
The song is full of sentiment and memory and love. And, in a way, that is what Winston is looking for. He wants to have real memories instead of fake ones like the ones he writes at work. He wants, in a way, to have love (and note that the woman sings this as Winston and Julia are meeting to make love).
So, the words of the song seem to me to reflect what Winston is trying to get out of life.
Winston knows that the song the woman sings as she hangs the laundry outside the window of the room above Mr. Charrington's flat has been composed by a machine, but he invests it and the woman singing it with a romantic pathos from the past, because, in his heart, he wants to believe that he has found a place of escape:
But the woman sang so tunefully as to turn the dreadful rubbish into an almost pleasant sound. He could hear the woman singing and the scrape of her shoes on the flagstones, and the cries of the children in the street . . .
His head tells him he is not safe, but the song, though manufactured by the Party, appeals to his heart:
It was only an ‘opeless fancy, / It passed like an Ipril dye, / But a look an’ a word an’ the dreams they stirred! / They ‘ave stolen my ‘eart awye!
The song's lyrics actually tell Winston what he doesn't want to believe. They are important for that reason, just as the lyrics of the Chestnut Tree song are important in explaining how Winston and Julia will betray each other.
The charwoman's song as she hangs the diapers tell Winston that his love for Julia and desire to escape into the past are "only a hopeless fancy." They will pass "like an April day," i.e., quickly. But looks and words—from Julia and O'Brien—have stirred up Winston's dreams and stolen his heart away. As the song states, Winston no longer operates from reason, but from imagination and dreams.
Winston responds emotionally and intuitively to the music and the way the woman sings, but blocks out what the words mean, mirroring the way he emotionally blocks out the reality of what he has done in opposing the state (though rationally he keeps reminding himself that none of this can work). He hears what he wants to hear.
Winston later listens to the woman singing over and over again and hanging diapers in inexhaustible amounts and imagines that the woman transgresses against the state by singing, calling it a "dangerous eccentricity."
The day Winston and Julia are arrested, the woman sings a different verse of the song:
‘They sye that time ‘eals all things, / They sye you can always forget; / But the smiles an’ the tears acrorss the years / They twist my ‘eart-strings yet!’
If Winston had been paying attention, he might have heard the foreshadowing in these words, suggesting that something bad is about to happen: he is being told that time heals all things and that he can forget—all hints that the good times in the room above the shop are about to end. But Winston can't hear the lyrics.
Instead, he idealizes the woman as "beautiful" and imagine proles like her leading the uprising against the state.
While we are never told explicitly that the woman works as a spy for the Party, the extent to which Orwell points repeatedly to how endlessly she stands in the yard hanging laundry and singing, hanging so many diapers that Winston has to create 30 grandchildren for her (though we never see any) would hint that she is, like Charrington, a tool of the state. Both she and the song she sings point to Winston's blindness and his desire to remake the world to conform to his own romantic image.