Over the years of stripping roofs when carrying out re-roofing the quality of the slates used then varied considerably from very poor square edge slates (no chamfer to the edges) to good quality welsh. But nearly all to some degree suffered from (nail sickness) this is where the nail holding the slate to the roof batten has failed (usually due to corrosion) poor quality galvanised nails being the main cause. Allowing the slate to slip out of place and cause roof leaks.
One solution to this, depending on the severity is to replace slates and re-secure, but this may be a false economy as the other slates may soon slip as well if the same batch of nails were used. This will entail on-going roof maintenance and the expense that goes with that.
A better option could be re-roofing, saving time hassle and in the long term expense.
Another problem that I have noticed on these roofs that do not have lead flashings but instead had a cement fillet detailing where the roof slope joins up with party walls is that there can be cracking where the cement fillet has come away from the wall allowing roof leaks to develop this is usually caused by the wooden roof structure settling over the years allowing the cement fillet to crack or detach from the party wall. Another reason to seriously consider re-roofing with the proper installation of lead flashings and underfelt as most of these roofs have none allowing loss of heat and drafts.