Alliteration in "London Snow" often supports the poem's themes of unification—for better and for worse. In a poem that's interested in how a busy, noisy city is brought together in shared delight, alliteration helps to create connections between unlike things. It often also mimics the physical qualities of the things it's used to describe.
For instance, in the first stanza of the poem, repeated soft sounds—/s/, /f/, /l/, /m/—help to evoke the softness, lightness, and muted nature of the snow. (Note that the words "soft," "light," and "muted" all use those very letters!) Take phrases like "flakes falling" and "Silently sifting." This pattern of soft-voiced alliteration reappears throughout the poem, reliably appearing wherever the snow is described.
Within all this softness, harder alliteration is used to describe the physical components of the city: for instance, in line 7, where "road, roof and railing" are all veiled by the snow. The repeated /r/ sound here contrasts with the gentle, quiet sounds attached to the snow, helping to establish both the city's literal and metaphorical roughness and to pull the city together as one conglomeration of human-made structures.
Alliteration also helps to give a shared identity to the "country company" (line 27), and to the laborers at the end of the poem:
trains of sombre men, past tale of number,
Tread long brown paths, as toward their toil
The hard repeated /t/ sounds mimic the repetitive rhythm of their trudge to work.