They are mostly part of the fae by all accounts. They werent present in SmF (that was the Gruffs, aka goat-people) but in CD they were mentioned both as part of the Winter Birthday Party as well as Lily's group of Summer at the Gardens.
Tolkien Dwarves were themselves based on the norse Svartalves, which have appeared several times, so you could say those are represented.
No direct mention of Gnomes in the text, though per wikipedia the term is only 5 centuries old, and "the term gnome became largely synonymous with other terms for "little people" by the 20th century, such as goblin, brownie" so one way to look at them is a broad category of Fae, of which we've seen several examples.