From 付(つ)け (tsuke, “連(れん)用(よう)形(けい) (ren'yōkei, “stem or continuative form”) of the verb 付(つ)ける (tsukeru).”) + 爪(つめ) (tsume). The tsume changes to zume as an instance of rendaku (連濁).
つけづめ • (tsukezume) ←つけづめ (tukedume)?