So, redesign all the clock watch-faces to be 24h instead of 12h too?

And replace the wristwatch of everyone who doesn't have a digital watch? And wall clocks?

Why redesign the watches? This problem doesn't exist in the places that use the 24h format (like most of the world around North America). We can use both time formats. I can either say 9 in the evening or 21 and everybody gets it.

Because "9 in the evening" doesn't mean much when you're a quarter of the world from GMT and forced to use GMT as your universal time.

Well, when someone asks me what time it is now, I don't say, "The minute hand of my watch is pointing at the 1, and the hour hand slightly past the 3." Instead I translate it to "three-oh-five". Wouldn't be too much of a stretch to say 15:05 instead.

(My point being that many watches and clocks already don't give us the literal time as it's spoken, just a representation.)

Edit: although just to be clear, I don't actually think we should switch to a single time zone. It would cause all kinds of headaches. I just think if in some universe it were to happen, it would necessarily go along with a switch to 24h time.

You can say 15:05 because you know it's the afternoon and so 3:05 translates to 15:05. Now think through how that works when the whole world is on a single timezone.

Yes, I agree that traveling would be hell with a single time zone regardless.