It was a very specific question and I know that you all like these kinds of specific challenges. I am willing to bet some of your fine minds have dove into ancient pedicure practices.
As this is more concerned with material culture and artifacts than with texts you might have better luck over at /r/askanthropology where the archaeologists are more likely to hang out.
Short answer: probably with a small knife, maybe with wear and tear, or with biting them (that last one probably not the first choice of an adult man, but it existed), maybe with pulling them off.
Long answer:
There are different ways to approach this question.
First of all, do we know that he would have cut them at all?
There was natural wear and tear, particularly for a man with a manual profession such as carpenter, walking around barefoot or in sandals. This likely played at least a partial role.
However, we know that people did not just rely on wear and tear to make their toenails shorter than giant claws for several reasons. There is logic of course, but also, there are some records.
When speaking about how to treat a female captive that a man wishes to marry, Deuteronomy 21:12 says to bring her into your home, wash her hair and trim her nails. Then content of Deuteronomy predates Jesus by almost 700 years, but it was still influential at his time.
We know that Jews still shortened their nails after the time of Jesus too, as the Babylonian Talmud (recorded about 400 years after Jesus, but with content based on earlier teachings), has instructions on what to do once they are cut, saying "the righteous bury their nails, the pious burn them, and the wicked carelessly discard them," lest pregnant women step on them and miscarry.
This site guesses at why this belief existed, including that cut nails were used in attempts at witchraft, but it does not appear to be historically rigorous so keep that in mind as you read it.
Rome governed everywhere that Jesus went during his lifetime, and we know that they also had ways of shortening nails.
They bit them, for one. The Roman satirist Horace described it about 35 BCE (Sat. I 10.71) when writing about a poet Lucilius that saying that while writing poetry he would often “as he wrought his verse he would often ... scratch his head and gnaw his nails to the quick” (uiuos et roderet unguis, Satires, Book 1), while Persius mocks what he says is bad poetry by saying it "tastes of bitten nails" (demorsos sapit unguis).
There are also Jewish records of tearing off and biting nails. The Mishnah, a codification of oral Jewish laws first assembled in the early 200s CE, specifically says, "one who pares his fingernails, either by means of his nails or by means of his teeth." This is in Mishnah VII. The Gemara, commentary on the Mishna created around 350-400 CE, comments on this, so we know that the nail reference was already there by then.
Still, that is biting fingernails more than cut toenails. And Romans had cut toenails. You can see it in their sculpture, such as this foot in Cesearea, built in what is now Israel in between 22 and 10/9 BCE.
You also see references to cut nails in their writing. For example, in the 100 CE, Artemidorus wrote that dreaming of cutting one's nails means that one will soon pay one's debts. Horace mocked a man as so miserly that he would cut his own nails in a barber's shop in around 35 BCE.
There are also records of nail cutting services. Wealthy people had a slave barber called a tonsor. For those who could not afford this, there were male and female public barbers who offered services including shaving, haircuts, trimming nail and also trimming corns. Diocletian's maximal edict of 301 BCE said that they should be paid two denarii for the service. This is not a lot, as it is also the amount that it gives a sheep shearer for food and care per sheep for a day. One hopes that this means that they were quick at their work and could get many customers.
But how did they (and possibly also Jesus) do it?
The use of small knives to cut nails is something found throughout history. For example, here is a picture of a razor (large round thing) and nail knife (small knife thing) from the European Hallstatt culture that flourished around 400-800 BCE.
The Romans specifically had a small knife called the cultellus tonsorius, or baber's small knife, that they used for many places where one might need a small knife. However, the Greek word for this knife was onychisterion lepton, which translates as light (small) nail trimmer.
The area where Jesus lived was Hellenized, and run by the Romans, they had these knives and, given that many people spoke a version of Greek called Koine. I wouldn't be surprised that they even called them a version of little nail cutting knives in Greek, but I do not actually know for sure the Koine word for that.
If you want to see a drawing of one of these knives found in Pompei, you can here. There is a paywall, but you can sign up for a free account that lets you see up to three articles at a time. That article, 'Tonsor Humanus': Razor and Toilet-Knife in Antiquity by George C. Boon is worth a read for more information on hygiene and self care at the time.
Knives are relatively easy to make, and one can always use a small knife, but another reason knives were used is because there was nothing else. Nail clippers are less than 200 years old. Even scissors were not an option - any scissors at all were very rare. There were shears for haircuts, but their construction did not allow fine enough cutting for nails.
So Jesus cut his toenails with a knife, and that he probably buried or burned his toenails once they were cut instead of throwing them somewhere, but do we know in which order he cut them?
Sadly, no. There are Jewish superstitions about in which order and on which days when one should cut nails, but the earliest references are about a thousand years after Jesus, and that is too much time to say with any sort of confidence that Jesus would have followed that practice, but I am ending with a mention because I think it is interesting.
And also because it is interesting, I will end on a note mentioning for how long biting was considered a legitimate option, at least for some people (possibly not for the feet of adult men doing manual labor in sandals or barefoot in hot climates). It is an early Soviet poster urging parents to cut their children's nails instead of just biting them. Young kids are a bit of a special case, as they are wiggly and using a knife would be tricky, but by that point, nail scissors were available, and the USSR wanted people to use them who were not.
See https://www.reddit.com/r/AskHistorians/wiki/health#wiki_trimming_nails
If you look at the Romans answers that may give a good answer. There is an example of a Roman Nail care kit at Vindolana fort in Northumbria England that I found quite interesting, but that was more a file and pick. But tiny shears were not out of the realms of possibility.
