Gamefreak logic is infamous for this kinda stuff. :3
Pokemon sometimes doesn't make sense so sometimes it's pointless to question it.
I ask myself sometimes how can Wooper and Shroomish can learn Punch Moves if they do not have hands?
I guess if you really want a reason, then the fact that we don't know Diglett's body underground may account for it. Maybe Diglett has a hand or claws or something that is not seen that allows it to use Cut etc., but then that would raise the question as to why you, as the trainer, don't see what's underground if Diglett uses it to Cut trees.
Also here is something very interesting I've found:
They are known, however, to have feet, as revealed in Pokémon Mystery Dungeon Explorers of Time and Explorers of Darkness and Pokémon Mystery Dungeon Explorers of Sky - Bulbapedia
Hope this helps.