Alliteration helps to give the reader the flavor of Amelia's friend's chatter. Working alongside the poem's bouncy meter, alliteration evokes the inescapable energy of the friend's conversation as she wonders at her old neighbor's transformation.
Take a look at how this works in the second stanza:
— "You left us in tatters, without shoes or socks,
Tired of digging potatoes, and spudding up docks;
And now you've gay bracelets and bright feathers three!" —
"Yes: that's how we dress when we're ruined," said she.
Alliteration works here to create that special feeling of being stopped in the street by a chatty old friend when perhaps you'd rather not be recognized: the repetition of sharp /t/ and blunt /d/ and /b/ sounds batters away at Amelia's facade of elegance. By contrast, Amelia's own speech is usually much softer—any alliteration that appears is typically on the gentle /w/ sound ("we [...] when we're").
Elsewhere, alliteration also helps to evoke the distance between Amelia's dismal past (and the friend's dismal present) and her fancy new life:
— "Your hands were like paws then, your face blue and bleak
But now I'm bewitched by your delicate cheek,
And your little gloves fit as on any la-dy!" —
Here, alliteration on /b/ shows the transition between Amelia's miserable "blue and bleak" past face and her "bewitching" present face, and the repetition of light /l/ sounds emphasizes her new ladylike delicacy. Again, Amelia's own speech sounds soft by comparison—characterized by those /w/ sounds:
"We never do work when we're ruined," said she.
The only time Amelia's speech gets a little feistier is at the very end of the poem, when she dips back into her country accent and her final two lines are marked by sharp alliteration of the hard /k/ sound:
"My dear — a raw country girl, such as you be,
Cannot quite expect that. [...]