This is a great question with no precise answer since each case can be different. The great folklore theoretician, Carl Wilhelm von Sydow (1878-1952), coined the term "memorate" to mean the accounts we tell about our own experiences. These are proto-legends, accounts of remarkable experiences we have had. The telling may or may not intentionally deviate from what we think really happened, and most memorates fail to circulate into third-party retellings of the account. Rather, they die with the first person. Some accounts can, however, begin the process of being retold, and in the process, they become polished, shedding details that don't work as well and gaining new details that make it a better story. This process can begin very quickly. Sometimes this process can be observed in the primary source record, and sometimes legends seem to leap out of nowhere with no documented origin (see my second comment).

I am conducting research into this process in a small central Iowa community. There was a local man who was famous for his accounts of his remarkable adventures (i.e. memorates), and many of these are now recounted second and third hand. Two days ago, I heard a story, which I collected from someone else in 1984. The incident occurred in the late 1930s, involving a person named Don B., an undertaker who died in 1981. His story about his exploit continues to be repeated in the community, and many of his other stories continue to be repeated. This account may or may not be faithful to actual details. No doubt the hardcore facts of the incident have mutated to enhance the story. We can call this a local legend by virtue of its being repeated by others:

This is about the time Don B. went to New Providence. Don was in the mortuary business there in Dixon, and New Providence was a little town about oh, eighteen, twenty mile from Dixon.

And he got a call that they had a corpse for him over there to pick up. A transient or a bum that had froze to death in a hay stack. So instead of taking the hearse, he just took the car, which was a big four door Buick sedan, and he got Kenny C. a guy there in Dixon who usually went with him on certain things.

So they drive over to New Providence. Turned out this guy had froze to death sitting in a hay stack. Sitting up. So they figured, “Well, what the heck. That will work all right.” So they just open the back door and set him in the back seat kind of in a sitting position. And they take off, heading back to Dixon.

There’s a young kid out there along the road some place hitch-hiking. So they stop and reach back to the back door, and he climbed in the back seat of the car with this guy who was dead. Sits down.

It was in the wintertime and just as cold as could be. This young kid, he turned to the guy, and he said, “Hey,” he said. “Cold, isn’t it?”

Of course, the guy didn’t say anything.

The kid tried two or three times to talk to this guy back there. And finally, he told Don up front, he said, “Hey,” he said. “This guy doesn’t talk much.”

And Don said, “Well, hell no. He’s dead! He can’t talk.”

And this guy bailed out of the car about fifty miles an hour. He left them right there.

I suppose there is a perception of the oral tradition being a thing of venerability, in the sense of age - though in reality it is just as reflective of a contemporary culture as anything else. So the answer may often though not always, in the case of oral tradition, be "very quickly".

Legends particularly would be tied to particular places and within the scope of known history (as opposed to Myths which tend to occur outside of time generally): Carl Lindahl, John McNamara and John Lindow's opening line to their definition, within their Medieval Folklore, serves as a decent rough guide:

A historicized and localized traditional oral prose narrative presented as a true account, often centering on a supernatural or other extraordinary occurrence.

So one would not be surprised to see legends about a historical event appear rather quickly in the oral tradition... I can think of numerous examples of this is modern culture of the past 20 years or so. The appearance of events within the mythic tradition is probably a much slower process however, though I would hesitate to be more specific than that. Within the Irish Oral Tradition, the Cromwellian Conquest takes on a theme that borders on myth in cases, and the same might be said for folk memory of the Famine - so there might be some aspect of particularly traumatic events that inspires this, but that is just idle speculation on my part.

I'm afraid Alexander falls outside my field, though I wouldn't be surprised for someone versed in the material to argue that the entire Western conception of Alexander is highly mythologized.

A second response involving the origin of legends - which indirectly points to a response to your question about how long the legendary process takes. Since we are generally unclear about where and how a legend forms, we are usually also unsure about how long that process takes.

The following is an excerpt from my Introduction to Folklore:

The answer to this question is somewhere between "we don't know" and "it's just what people do" – that is, they invent these sorts of stories. Several folklorists, including the late Alan Dundes and Jan Harold Brunvand, recognized an opportunity with recent urban legends to finally track down "the source," the point of creation, to better understand how stories originate. For the most part, they were unable to find the point/person of origin to any general degree of satisfaction. It is a maddening question, and much like the proverbial chicken and the egg, it isn't likely to be resolved.

People tell stories. People repeat stories. And as they repeat them, people modify stories, which sometimes results in the birth of new stories. There might have been some highly imaginative, creative people who invented stories at some point, but that sort of person has yet to be identified in the annals of folklore studies. Folklorists often asked renowned storytellers to tell a new story, to invent something that they had not heard. Without exception, these requests were refused by the storytellers who explained that while they repeated stories, they did not invent them. In general, it may be best to consider the possibility that the process of repetition and transmission is the source of much of the creativity that is evidenced in the international library of oral tradition.

One example may help shed light on the creative process: the Cornish droll tellers were known for their wild adaptation of the material they heard. Their counterparts in Ireland were the seanchaithe, professional storytellers who prided themselves in the faithful recitation of stories they had heard, as close to the original as possible. Perhaps in this, we can see the opportunity for new stories to be born, and indeed, Cornish oral tradition includes more than its fair share of distinct subtypes (i.e. variants) of legends and folktales, perhaps offering a hint as to how these things may come into existence.