When you fight Letho in Chapter 1 he knocks your sword from your hand and after the cutscene you have to walk over and pick it up again. Do you think you could have lost it there or do you remember having it after that?

In either case I would visit the nearest camp and start checking the vendors for weapons. Any decent steel sword will probably be better than silver for human opponents. Most camps will have a blacksmith that will sell basic weapons and usually has diagrams to create more advanced swords.

Yeah, well, that's the thing, I had the sword, but it disappeared from my inventory yet Geralt still had it equipped and could use it somehow, but as soon as I equipped something else there I'd lose the sword, I have no idea what would have happened if I had to deal with humans or elves in this condition, but I managed to find a brand new sword before that happened.
So everything's fine now.

When you don't have your own steel sword equipped, the game will default to a base steel sword that you use so you're never without a steel sword.