do whatever it takes to fix it, it's YOUR investment too. don't wait for the neighbors to do the right thing. Feel for you.

Did the roofer say what a permanent fix would be?

So the neighbors extension was just built?

Ooops didn't see the pics/video posted. That is some mess.

IMO who built the house , is it a new build . I need more info for sure and you need to fix what ever is happening to your house . Where I live this would never happen since our homes are much further apart and I need to see pics from much further back and honestly i don't know what oyu are showing in those pics.

It doesn't look like there is an adequate solution for you. There's probably some issues and responsibilities about who owns what and which wall, it looks like 3 coming together here.

The gap looks to be about 2" wide, too small to get anything in there or down in between. You can see the collection of all the debris washing down from their tiled shingles, if not already there will be so much organic material collected in that gap down at the ground it will retain moisture and that will translate through your wall. And stink.

Besides, no matter what you do, their pitched roof will still route water and debris up against your vertical wall. Their roof needs a cricket built to route the water the other (perpendicular) way.

As a temporary stop gap, you "could" put a flashing tucked under their 2nd shingle and up your wall. However, the water is still coming straight to the wall, so it will backup under their shingles and cause lots more trouble for both of you.

I'd start by cleaning the debris out of the gap. I suspect it's acting like a wet sponge and holding the moisture against your wall.

What is the separation between your two walls below the roofs and what are they constructed of?

Looks like some flashing is missing.