Here's the second stanza of the woman's song:
They sye that time 'eals all things,
They sye you can always forget;
But the smiles an' the tears across the years
They twist my 'eart-strings yet!
Orwell gives us the analysis himself in his narration:
She knew the whole drivelling song by heart, it seemed. Her voice floated upward with the sweet summer air, very tuneful, charged with a sort of happy melancholy. One had the feeling that she would have been perfectly content, if the June evening had been endless and the supply of clothes inexhaustible, to remain there for a thousand years, pegging out diapers and singing rubbish. It struck him as a curious fact that he had never heard a member of the Party singing alone and spontaneously. It would even have seemed slightly unorthodox, a dangerous eccentricity, like talking to oneself. Perhaps it was only when people were somewhere near the starvation level that they had anything to sing about.
The significance of the song is not the lyrics; it's the fact that she sang "alone and spontaneously." Whereas the Party members recite nationalistic verses in unison and without feeling, the woman sings a folk song, a kind of blues that aches with comic and tragic feeling.
Psychoanalytically, a woman hanging diapers must remind Winston of his mother, whom he lost. Her lyrics about "time heals" and "never forget" and "smiles and tears across the years" reveal his desire to be nurtured and loved. There, with Julia, must be the first time in years that's he's felt that safe and vulnerable, so her song triggers his memory of childhood.