A few things to try:
1. Put the edge of a credit card or other plastic card on each set of three frets (moving up one fret at a time) and see if you can rock the credit card due to uneven frets. Do this on both the bass and treble sides of the neck.
Watch the ukulele from the side of the neck to see which fret the string touches when buzzing, and test that fret and and its neighbors four times, once next to each string.
2. Put the headstock of the ukulele towards your eye and look down the side of the neck (do each side of the neck). It should be slighly concave (called “relief”), but have no irregularities like bumps or wavey spots.
3. Put new strings on. A string that hasn’t stretched evenly or has some manufacturing flaw making it not quite the same along its length can produce all sorts of weirdness. So if your problem is specific to one string, replacing the string is probably worth a try.
4. Tune the ukulele down 1-2 full steps to reduce string tension and make it easier to fret. Raising the action can be good when multiple frets are buzzing, but when it’s just one fret then low action probably isn’t the problem. And having the action too high can make it harder to fret properly, which may show up as buzzes at spots where your neck/frets were slightly quirky and proper fretting was particularly necessary. So if the buzz goes away when you lower the string tension, consider *lowering* the action, perhaps especially at the nut end.
5. Try using a capo, moving it up one fret at a time and playing the open strings, and see if you get the buzz in the same place you would while fretting. (If it does, your fretting technique is definitely not the problem.) For this test to be valid, your capo should not be adding new buzzes at places where you don’t normally get them - if it does, try adjusting capo placement or pressure. (If you are using a guitar capo, lower the capo pressure before you start, a nylon-string ukulele doesn’t need nearly as much pressure.)
6. Then put the capo on the first fret, and try fretting the problem spot with your fingers. If you can fret without a buzz while the capo is at the first fret, high action at the nut end may be making it hard for you to fret properly when the capo is off, and that may show up as increased risk of buzz anywhere your neck/frets have slight quirks that make proper fretting particularly important.
(If you do not have a capo, test by fretting with two fingers on the same string - one for the fret before the one you want to play, and one for the fret you want to play.)
7. Try fretting at the problem spot with a different finger than you would normally use, and see if the buzz goes away or changes. Press hard with your middle or index finger, press lightly with your pinky finger, try all sorts of combinations and observe if the buzz changes in response. Also try varying how close your finger is to the fret - usually it’s best to have your finger close to the fret, but occasionally if you’re fighting high action fretting *slightly* further back from the fret may be a little easier.
8. Try playing a different ukulele of the same size. If the buzz follows you across instruments, pay very close attention to your technique and what you’re doing when the buzz happens. Or have someone else try playing your ukulele and see if they get the same buzz at the same spot.
(This list is mostly a combination of advice I received from a few sources (a luthier and a family member who teaches guitar) when I was having buzz issues. I make no claims to being an expert on this subject, I’m just passing along what people I trust told me.)
Note: Most of this list assumes the buzz happens while you are fretting. If the buzz happens even while playing an open string, focus on steps 1-3.
