Does she have a relationship with the neighbour? If she does, she can enlist her help to work on systematic desensitisation, starting with hanging out something small like a sock and building it up.
It may help to have the dog out in the garden while the washing is being hung up (again, working within the dog's threshold).
If there's no cooperation, I'd use the same "pressure on/pressure off" protocol that I've used extensively with Willow for leaving the house, getting in the car etc, where I work within the dog's limits to eat pretty boring food (eg kibble) and toss one piece in the direction of the scary thing and then one away from the scary thing. If the dog won't go for the "towards" one, back it up and try at a greater distance. The food should be boring because you don't want the dog pushing themselves into the "gagh" zone just for something especially tasty. The tossing the treat away gives the dog some relief and also gets them into a rhythm where they are just ping-ponging back and forth. If the dog doesn't take a "toward" treat, immediately throw one away and then on the next rep make it easier. Don't try to coax the dog into crossing their comfort level. Their tolerance will grow faster from not being pushed.