Hi, and welcome.
Unfortunately, your cat is a pretty tame little kitty, because the question as asked isn't specific enough.
If by trad you mean Irish trad, Scottish trad, etc., then clearly not, prima facie
. It's not generally recognized by players themselves as a standard part of the Irish trad instrumentarium, except to the extent that any instrument may occasionally turn up.
If you mean Hawai'ian trad, then I'd say probably yes, to the extent it's widely perceived as such and has evolved from its Portuguese roots into an emblematic sound in the folk and pop music of Hawai'i.
If you mean: Is the ukulele capable of performing Irish trad repertoire, then clearly yes. Both the fingerstyle and rhythmic strum approaches of Irish trad guitar translate fairly directly to ukulele, accounting for only 4 strings and reduced compass. And the reentrant tuning of the higher ukes combined with non-linear "harp" fingerings makes possible a nice simulation of some Irish harping techniques.
Key can be a problem for ensemble and backup playing, as the standard C tuning makes the uke effectively a transposing instrument in F. But that can be approached by capoing, D tuning, translating the IrTrad chord patterns, or a combination of all the above. Guitarists who play in the prevailing Irish tunings of DADGAD and/or Dropped-D may also need to deal with retuning, or translating back to standard tuning.
If you mean: Could the ukulele enter the Irish tradition as guitar and bouzouki have done in fairly recent times? Maybe, but I tend to think not, except for idiosyncratic use. Some of the issues cited above are obstacles. The ground of percussive stringy things is already well-covered, and still not accepted by everyone. And like nylon strung guitar, nylon strung uke is volume-challenged when it comes to meeting backup expectations except in very intimate or amplified settings.
On the other hand, the current surge in ukulele revival interest does parallel the pop interest in banjo and guitar before their entry into Irish trad. So who knows?
Are you interested in Irish trad yourself? Or other kinds of traditional music?