Ooh I love this stuff! So, to begin with, I'll link you to my (/u/telekineticm) post here where I discuss the historical accuracy of the Odyssey. In that post, I mention that prior to Heinrich Schliemann, it was unknown if the city of Troy even existed. Of course, we do now know that Troy existed as a city, but it was destroyed ~1200 BCE, about 400 years before the Homer mythos emerges. Because really, the Troy mythos and the Homer mythos, while intertwined, are not the same thing. The Iliad itself was composed by a variety of poets, each who added to the story while "ghostwriting" under the name of Homer. This article uses the catalogue of ships in the Iliad as proof that some sort of Iliad-esque ware likely did happen.

I occasionally see this tumblr post about how the creation of the Iliad probably resembled a D and D game, which is actually pretty accurate. Another good analogy would be fanfiction--the story of the Trojan War is based on the extant myths of the era (Zeus et al.), but the Iliad itself was added to by each teller, as mentioned above. So, we can trace these stories back to ~800, since that is when "Homer" supposedly lived, but because these texts were not written, it is difficult to definitively trace this narrative any further back than this era. However, it is possible to "deduce [. . .] that behind the specifically Homeric epics there lies a long 'prehistory' of hexameter heroic songs and cycles of songs which feed directly into Homeric epic, and which certainly goes back as far as the mid-second millennium, and probably a great deal further" (article here). This deduction is based on analysis of Homeric linguistic patterns* and references to material culture that are archaeologically known to be from as early as the proto-palatial Mycenaean era.

So we know that the oral tradition we call Homeric has been going on for millennia. But it is unclear when the idea of "The Trojan War ^tm " came into existence. Like any other form of media, oral traditions are alive, and being influenced by popular culture and even intellectual trends of the era, is a key part of the oral tradition. So each reteller of the story adds their own trope or aspect or lens or even local folktales into the larger epic.

So the short answer to your question is that as far as can be proven, the Trojan War as a unified narrative, with all its accessory and component narratives, dates from ~800 BCE. However, the tradition that Homer is a part of can be traced much further back into history than that.

*If you are interested in the history and composition of oral traditions, particularly with regards to Homer, here is an article that compares the Homeric tradition to the extant Albanian oral poetry tradition--it's an excellent introduction to the study of oral traditions and really breaks down the peritactic composition of oral poetry in a comprehensible way.