Learn Hardcore Turkish: Yapı ruhsatının gecikmesi durumunda iş programı değişecek. - If the building permit is delayed, the work schedule will change.

Because the sentence uses a noun-based clause: yapı ruhsatının gecikmesi = the building permit’s delay / the building permit being delayed.

In this structure, the thing experiencing the action goes into the genitive case:

yapı ruhsatı = building permit
yapı ruhsatının = of the building permit

So -nın marks ruhsatı as the possessor/subject of the nominalized action gecikmesi.

A very common Turkish pattern is:

X-in Y-mesi = X’s doing Y / X being Y-ed

Here:

yapı ruhsatının gecikmesi = the building permit’s being delayed

What exactly is gecikmesi?

Gecikmesi comes from the verb gecikmek, which means to be delayed or to be late.

It breaks down like this:

gecik- = delay / be delayed
-me = verbal noun / nominalizer
-si = third-person possessive ending

So gecikmesi literally means something like:

its delay
its being delayed

In this sentence, it does not mean a normal possession like his book. It is part of a standard Turkish grammar pattern where a verb is turned into a noun phrase.