You did not specify how technologically advanced this species is. They wear clothing under water, which means that is produced somehow. Presumably underwater as having things made on the surface for import seems like it would be world breaking. Hands, with watches, but no mention of wrists or what exactly those hands are and how they function. It seems like the watch is the least of your worries so far.
So off the bat we have some obvious hurdles that may demand more explanation than a watch. But let's suppose the under water peoples have managed to escalate the realms of their available technology to a point where we might as well be making up far out science fiction beyond the realms of plausibility in our own world. That much we can call fair game.
Why not use imposed gesture based holographics then? Who knows how they managed to isolate electronic tech under water safely, but let's suppose this tech has some kind of hermetical barrier that allows these beings to interface without the threat of obvious shortages and physics and chemistry based conflicts. Why not just have them turn their flippers inward and a holographic, gesture based watch appears before them? Who cares about the hows and whys of this anymore since we're already way off anything we might conceivably see on our own world?
Our own smart watches keep their screens off until we gesture already. This tech is low level by this point. So perhaps it is not beyond plausible that some being developed something like a photonic resonance that projects illumination from something more static, like a badge, that acts as the interpreter for all of the beings' computing needs. Think of it like a relay. Turn the flipper inward, think "time" and a watch appears over the hand-ish thing. Same for things like opening locks or whatever else is going on in your underwater realms. Centrally based, possibly non-submerged computers relay to the badges and project what they need to the user. Handwavium up the a** but you're going to need to adopt that as a necessity for anything from our world to exist in this one.
Officially, I always suggest people avoid over explanation. If you need a watch, just give them one. Like a barnacle based adhesive outer garment pseudo-pocket watch thing. If the watch is what flips your story from believable to unbelievable, then there are problems elsewhere already.
You might benefit from incorporating something along the lines of "alexa, what time is it" and just have a phantom voice answer in whatever means of communicating these beings use. That would at least be relatable.