It's impossible to know from the info we have that the searches weren't completed properly. If there is a deed that isn't referred to on the title when it should have been, and the seller didn't provide a copy, how is the conveyancer supposed to know about it? Searches aren't physical; they don't involve going to the neighbour's house and rifling through his desk for old deeds!
If it should have been registered and wasn't, the fault lies with whomever was due to register it. That may well mean the OP isn't bound by it, see the article I posted above. If the neighbour then wants to pursue whomever should have registered his lost right, that's up to him.
Similarly with an ongoing dispute - if the conveyancer asked for details of any dispute (which is in the standard docs anyway), he isn't a psychic. If the seller concealed a dispute, he's at fault for misrepresenting the position, not the conveyancer.
If it's an equitable right, it may well not be registered/registrable and only a lawyer can tell you after asking OP questions and reviewing the file.
Blanket statements like "no such thing as washing rights" are dangerous because it's not true. If you specifically allow your neighbour to hang washing on your land, that's a right. If your neighbour uses your land to dry his washing (or park on your drive, or run something over your land, or walk across your garden etc etc) for a long period of time without being challenged, he can acquire permanent rights to do it. Every case will be different and will depend on its own facts.
Here, if there is a written right and it wasn't registered when it ought to have been, the most simple thing to do would be to say that OP bought the house for a fair price and therefore is not bound by it. See if the neighbour puts his hand in his pocket to take it to court after that. But here the OP was advised to have the right put on her deeds, so this option may now have gone. She needs to check why this advice was given and whether it's negligent, in addition to everything else said above. After all, such a right may well affect saleability and price, so it should have been reviewed carefully.