Hm... I've seen this claim and the related claim that popping pimples in that area can cause fatal infections several times.

My first note is that I've never seen someone claim trigeminal nerve infection is the mechanism, and indeed trigeminal nerve infections appear to be exceedingly rare and not directly life-threatening. The relevant disorder would be trigeminal neuralgia, which causes severe and lasting pain, but is almost always a product of compression, inflammation, or demyelination of the nerve. I can't find any record of infection as a cause, only some hypothesizing that atypical forms of the disease might be based on infection. The only major connection I can find between hairplucking and that nerve is the claim that it's responsible for the sneezing reflex many people have when nose hairs are pulled.

As for "can plucking nose hairs kill you?" That's a bit more credible.

The mechanism here isn't nerve infection, it's infection of the veins in the area, which have unusually direct connections to the cavernous sinus and brain. This can lead to meningitis, brain abscesses, and disabling/fatal bloodclots, so I suspect it's the same thing described above. (It's worth noting that normal "sinus infections" can also have the same effect, but the direct bloodflow connection here threatens much faster escalation than normal sinusitis.)

On to rates, then. Meningitis, brain abscess, and cavernous sinus thrombosis (a blood clot in the cavernous sinus) are all exceedingly dangerous, so let's just worry about the rate of getting there.

The base rate of cavernous venous thrombosis is ~1 per 100,000 per year, mostly occurring alongside other major facial infections. Without strong reason to expect that you have heightened risk, it's largely irrelevant.

A 1937 study found that 61% of cases of cavernous sinus thrombosis (which is extremely dangerous) were a product of upper-face boils. Modern rates appear to be closer to 50% of cases, probably down to a mixture of dental surgery causing new cases and antibiotics preventing nasal-infection cases. (A venous infection progressing straight to brain abscess or meningitis appears to be possible, but unlikely without an intermediate infected-sinus stage.) Boils around the middle of the face are viewed as deserving prompt intervention to avoid this outcome.

Unfortunately, there's no clear data on how many of those cases were the result of hair-plucking. Staph infection can lead to boils at any irritated skin site, and are indeed the infectious agent in 70% of thrombosis cases. But severe boils are vastly more common in patients with specific risk factors (family history, immunosuppression, certain disease). Indeed most people who pluck hair never seem to develop boils. (N.b. boils are not the same as 'clean' inflammations from trapped hairs.) At-risk patients are also in far more danger from boils, since they face risks like immunosuppression and frequent treatment leading to drug resistance. One-time cases in healthy patients can be treated simply and effectively with antibiotics, steroids, and lancing.

So: plucking can lead to boils, and boils can lead to thrombosis, and thrombosis can lead to death. But any major infection from plucking appears to come after a noticeable period of surface infection at the site (i.e. a boil). Simple one-off boils are extremely treatable in the presence of modern antibiotics, so the risk can be arrested almost completely at that point as long as you seek prompt medical treatment. (Go to the doctor if you have a boil anywhere anyway, because they rarely heal without treatment.)

As far as I can tell, this is one of those risks which is discussed because it's interesting and logical more than because it actually hurts anyone. "Don't pluck nose hairs" does not seem to be an important precaution unless you have other major risk factors like HIV or a drug-resistant staph infection.

https://en.wikipedia.org/wiki/Danger_triangle_of_the_face

https://www.ncbi.nlm.nih.gov/books/NBK448177/

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1390530/