If it's a downward or sideways stroke that cut the nose off, that probably wouldn't be fatal, assuming someone at least staunched the blood flow with a wad of cloth. The immediate danger would be fainting/shock, which could be made worse if the weather is bad or the person ill-nourished. After that infection would become the danger. If the injury healed, that would be it, no further serious problems (well, maybe depression from looking horrible). Minor problems would include difficulties with swimming (water in the sinuses) and cold weather.
But if you're talking about the sword piercing into the brain cavity, that would probably be fatal. That kind of cut would also probably blind the victim in one or both eyes.
If you want other kinds of fatal facial injuries, people have been killed by a strong blow to the nose shoving the nasal bone up into the brain. And stabbing through an eye into the brain is a traditional way to kill big armored monsters, and works on humans too.