Hi Bibs,
I am into roofing but more into industrial metal roofing. I have quite a bit of domestic roofing experience.
So here's what I can see:-
1. The new lead extends across the entire width of chimney breast. This is a good thing as they can't just stop on the boundary otherwise it would leak where they stopped.
2. Ridge tiles joining the two properties is a new one on me and not really good practice. They should have either used lead or a GRP secret gutter detail.
3. The two tiles on your sister's side right next to lead gutter aren't positioned correctly. They should be like the tiles on the adjacent property. You cand see how much it's short by the organe side lap that the tiles used to cover near corner.
4. There's a corner off the secoond tile up from gutter near ridge tiles. It will be leaking there!
5. Without stripping the tiles off above lead gutter, the lead gutter could be short and not lapped under roof tiles enough. It should go upto roughly where the second tiles are. There's no science behind exact measurements but you need a reasonable depth in the lead gutter in order to get water away before it back up and over the back of the lead gutter.
6. The pointing is poor and hasn't been angled off for water to drip off. This is not likely to be cause, more damp over time.
So I would say remove a couple of the ridge tiles (held in place by cement) and lift up tow rows of tiles. You'll see your problem then. It will be either the two tiles need sliding down, the broken tile on second row or the lead gutter not having enough distance under the tiles. It will take about 2-4 hours labour to do this work providing it doesn't need a new lead gutter. If it does then add another 4 hours.
I hope this helps and if you need any more info, ring me anytime!