Though the poem doesn't have a clear rhyme scheme, its language is still quite musical to the ear. This is in part because of alliteration, which appears relatively consistently throughout the poem, though often in a restrained, subtle way.
Often, the poem's alliteration echoes the content of the lines in which it appears. For example, in the first line, there is clear alliteration of the /l/ sound in “long” and “ladder.” This repeated, leisurely sound seems to almost lengthen the phrase itself. By contrast, the influx of /s/, /t/, and /st/ sounds in lines 1 and 2 pop out sharply, almost as if they're "sticking through" the line just as the ladder is "sticking through a tree toward heaven still."
There is also frequently alliteration of /a/ sounds, perhaps unsurprising given the title of the poem (which is itself alliterative: "After Apple-Picking"). This subtly evokes the presence of apples throughout the poem, perhaps reflecting how the speaker can't stop thinking about them.
Alliteration can also suggest a sense of fatigue, as the reader's ear may grow tired of hearing the same sound over and over. Note the alliteration (created in part through phrasal repetition) of /o/, /l/, and /h/ sounds line 26 and 27:
Of load on load of apples coming in.
For I have had too much
There is also further consonance /l/ and assonance of /o/ sounds here, which, combined with the alliteration, create a sense of utter exhaustion. The long /o/ is like a repeated yawn, while the /l/ slows down the line itself. No wonder the speaker has had "too much" of this!