Yes, all the previous posters are quite correct in that your neighbour doesn't have to dispose of the lopped branches herself and the law is quite clear that lopped branches have to be returned to the legal owner of the tree. Why should she have to go to the bother of clearing away rubbish which doesn't belong to her?
I'm sure having the cuttings strewn all over his garden is annoying to your father, but it's probably equally annoying to his neighbour having them intrude over her fence, invading her space and shutting out her sunlight. And if you're father is irate about it, I'd be tempted to do the same and keep out of his way too !
Seriously, the only way for your dad to avoid this hassle is to take responsibility for the trees and get somebody in on an annual basis to prune them back and take away all the rubbish.