Dave has summed it up pretty well. Looking at the new lead work it is typical of how you will often see chimney flashings; see page 33 of this lead flashing guide and, in particular, the bossed apron, side flashing and bossed back gutter.
https://www.associatedlead.co.uk/wp-content/uploads/2018/06/Guide-to-Rolled-Lead.pdf
The main question for me would be how far is the back gutter lapped under the tiles as this cannot be determined from the photographs.
From the internal photograph the back gutter doesn't appear to have proper support as the timber boarding looks rotten / missing. As has been said above, the condition of the rafters needs investigating and the sarking felt is perishing too.
It is a pity that the neighbour or roofer didn't approach your sister and explain what was going to be done and ask if she wanted her side done too as her flashings do not look to be in good condition and the side flashing looks like it may have been covered with 'flashband' or something similar (it's hard to tell).