yjones, good evening again.
Images posted above assist a lot, they can be enhanced and brightened up as well.
Several things.
1/. in the last posted image, there is a lot of "vegetation" [ferns] growing out of the mortar joints, never a great sighn, the ferns need to be totally removed?
2/. I the first posted image, there appears to be a lot of missing mortar on the right hand side near the upper [back] end of the side of the chimney, the position of the missing mortar is close to the line of the lead flashing, this will allow rain water an entry point?
3/. In image one posted there may be a load of debris built up between the lead and the slate ?
4/. The images posted appear to indicate that there may??? be two layers of lead, one stepped. the other may?? have been run over the top of the stepped lead flashing?? as seen in images 2 + 3 this could indicate historic problems with the flashing's, hence the need to insert the second flashing over the first??
5/. The cap, the "topping" to the stack appears to be thin? could be a skim of mortar only, this can crack and admit rain?
6/. There appears to be lead wedges that can be seen at the top of the lead, where it would appear that the lead has indeed been tucked into a raggle formed in the brick [the wedges jamb the lead into place to assist in holding the flashing's in position] however, some pointing appears to be missing at this raggle.
In general terms an overhaul of the stack would not go amiss, a general good rake out and re-point, at the same time a good look at the flashing's would be a necessity
As for costs?? depends where you live in the UK
Ken