The lead work does look wider than 6", probably 9" or more. Going up the lay board would probably require more like 18" as mentioned above.
The three bottom ridge do look to have been re-bedded fairly recently as they are on a slightly higher morter bed and you can see a lighter grey one that has been replaced with similar as on the neighbours roof. This may have been already broken and be the cause of the ingress to start with as there are signs of damp higher up the rafter (also causing the neighbours problem by travelling along the trimmer that will stetch into both properties) or it broke when they lifted the ridge. Similar as to what may have happened on the lower ridge where two tiles have been replaced on the neighbours side and one on your sister's.
They may have used a wider lead on the lay board, you can't tell unless you lift the bottom tile. Some terrace roofs have the coverings run continuosly along the length of all the buildings but sometimes you may have the party wall extend above the roof line with a capping of some sort. Here seems to be a ridge capping, are they all like this along the street?
If they have lifted the three ridge tiles I would expect that this would have been done to make good any damged felt. It's possible this has not been done on both sides as only instucted to carry out work on one side. Although informing the neighbour would be courteous.
Try getting their contact details and ask them of their findings whilst they were up there. 🙂