Truthfully, we don't know
There are not very many details given about Narsil other then that Elendil wielded it and that it was forged by Telchar. It is not known when Telchar forged it either. While the Dragon-helm was forged sometime before 471, it is not certain that Narsil was forged then, around the 4th century. Although it was also around this time that Angrist (another of Telchar's inventions) was forged. It is possible it was commissioned by Thingol in 1330 (in the Years of the Trees), this would however mean that Telchar had lived for over 600 years (assuming that the Dragon-helm of Dorlomin was produced around the 4th or 5th centuries).
Therefore Thingol bethought [him] of arms, which before his folk had not needed, and these at first the Naugrim smithied for him. For they were greatly skilled in such work, though none among them surpassed the craftsmen of Nogrod, of whom Telchar the Smith was the greatest in renown.
War of the Jewels, The Grey Annals
While it's possible the narrator could just be stating Telchar was the greatest in renown this would be strange to state in the annals of a history before he had lived.
If we assume this is true, the sword must've somehow passed from Thingol into the hands of Elendil. The most likely pathway for this would've been through the Elros, first king of Westernesse. While there is no evidence that this is the path it took it is possible that like Aranrúth, the sword of Thingol, Narsil had passed to Elros. Unlike Aranrúth, however, the sword must've passed down the first born line as opposed to the first born male line to reach Elendil. This again seems possible given at the time Narsil had remained unnamed and would not have been preferred as an heirloom fit for Kings over the sword of Thingol.
The King and most of the great chieftains possessed swords as heirlooms of their fathers...
Unfinished Tales, Part 2, Chapter 1: Description of the Island of Númenor
Although throughout all the above speculation we assume the sword was of any notoriety before it. The earliest (in-universe, I believe) mention of its name, however, only comes around the Battle of Dagorlad:
...and the sword of Elendil filled Orcs and Men with fear, for it shone with the light of the sun and of the moon, and it was named Narsil.
The Silmarillion, Of the Rings of Power and the Third Age
Conclusion and summary of canon knowledge
As far as is canonical, we know that Telchar forged it sometime between The late Years of the Trees and the end of the First Age (most likely). It somehow made its ways from the hands of the Dwarves of Nogrod or the Elves of Beleriand into the hands of Elendil. It was only named Narsil after Elendil bore it during the Battle of Dagorlad.
Predictions made using knowledge from canon
- It is possible it was in the armouries of Beleriand and owned by Thingol.
- If so, it is possible it was passed to Elros, King of the Men of Númenor, after the fall.
- It may have passed down the first-born line to Elendil as it was "lesser" to Aranrúth, Thingol's personal sword.
- And through that line came into the hands of Elendil along with the Palantír, the Sceptre of Annúminas, the Ring of Barahir and a seedling of Nimloth.
