We have a "flat roof" (actually a very slight pitch) which is made from lead with folded seams.

It is rectangular, with a slight slope toward one long side where there is a gutter with a downspout at either end.

We had our first proper rain in a while today, and it was torrential (and is still going, and forecast to be continuous for the next 24 hours). I quickly discovered that one of the aforementioned downspouts was blocked, and that the blockage was under the footpath between the bottom of the downspout and the outlet to the street gutter, so I detached the bottom of the downspout so that the water could freely run onto the footpath.

There are several leaks into the rooms underneath the flat roof. I think these are exacerbated by the fact that the downspout was blocked, but obviously they are not caused by that.

How can I find the cause of leaks in this roof? Some of the seams have been flattened by people walking on them over the years -- would that cause leaks? There are no other obvious signs of damage, but I'm not sure what I'm looking for.