Well I guess it must have something to do with the tripped breaker at the neighbour's house. Easy fix, just tell the neighbour that they can never turn that breaker on ever again, lol. Just kidding.
You say that when their breaker was tripped you had no voltage. Then when you reset it you had voltage from the downspout to the cold water pipe.
Did the neighbour's hot from that circuit come into contact with the downspout or something else that is bonded to it? Nail or something?