First and most importantly, take the water away from between the properties right , even if it is with an old length of open guttering, used as a channel.
Call the people who fitted your gutters and facia, they will come back and repair what is probably a simple job and should be on warranty.
This will cheer up your neighbour, so tell them the good news, have a chat with them to pull their builder back and fix the downpipe problem (they may not even know about it) as both properties WILL be damaged in no time.
Good luck