If I put a woodchopping block in the garden of my house and try to use it without an axe, I get the message telling me I need an axe, but the character goes through the woodchopping animation anyway. I've tried copying one from Riverwoood but the same thing happens. If I actually use the one in Riverwood it works correctly.

I can't see anything else in Riverwood apart from the block to copy. Anyone have an idea?

Sounds like the regular activation is blocked ( BlockActivation() ) beforehand in another script.