Alliteration appears in almost every line of the poem, filling it with music. Much of this alliteration actually echoes the alliteration that appears in Marlowe's poem, "The Passionate Shepherd to His Love," to which this poem responds. In a way, then, alliteration is a key tool with which this poem mocks the original.
For example, "pretty pleasures" in this poem picks up on the /p/ sound in "pleasures prove" from Marlowe's. That crisp sound also makes the phrase itself seem "pretty" and "pleasant"—perhaps self-consciously so.
While Marlowe's poem uses alliteration to suggest, beauty, natural abundance, and carefree joy, the nymph flips this on its head. In her poem, alliteration becomes part of her rejection of his offer. Here is the second stanza, for example:
Time drives the flocks from field to fold,
When Rivers rage and Rocks grow cold,
And Philomel becometh dumb,
The rest complains of cares to come.
The alliteration here is showy and over-the-top, firmly deriding the shepherd's own poetic attempts. The /f/ sounds make the flocks' march from "field to fold"—which stands in for the onset of winter and, ultimately, death—seem inevitable. The gritty /r/ and sharp /c/ come across as violent and threatening, a far cry away from the idyllic vision of the countryside offered by the shepherd.
The nymph saves some alliteration for her final rejection at the end of the poem. The four /m/ sounds in "my mind might move" make her "no" emphatic and final, showing that her mind is firmly made up.