Have followed with interest the RCD (etc) discussions and have no comments there due to my limited knowledge.
Assuming a RCD problem can be sorted as being a faulty unit, then perhaps there is another way of looking at the original problem?
The OP's description of neighbour needs a bit more clarification. Is this a semi or terrace house? If so, is it possible that there has been a connection made between the two via, say, a loft light, aerial, outside light, cellar or through the dividing wall? There would still need to be a fault of course, but it could be simple to find (?)
When the electrician called did he look just at the OP's house or call at the neighbours?
My house is the end of a terrace of three houses. It is hard to determine exactly how the overhead power cables are connected to the houses but there are two big thick ones coming into my property, via the neighbour's house. I can't clarify more easily since it's difficult to see what is happening - the cable appears to run along the wall and then split off down into his house as a single cable but continue on into my house as two cables.
The electrician didn't do a great deal of investigation. He had a look next door and said the neighbour has been converted at some point and only has one cable going in, but I'm not sure what this is about. Sorry