You asked two questions...
What if there were only one time zone?
As a historical reminder, Time Zones are a relatively recent invention, beginning basically in the 19th century, thanks in large part to the widespread use of the railroad.
As others have stated, China only has one time zone, and Spain has up to a 2.5-hour lag between sunset times at east vs. west ends of the country. Groups like the US Military, air travel, and others prefer to use a single, agreed-upon time like GMT, to reduce the possibility of time-zone confusion. Also, people are relatively used to working in non-standard (eg 9 to 5) times, given the prevalence of 24-hour operations or jobs that require frequent travel. This proves it can be done.
There would be some confusion during the transition phase between time zones and time zone. But people would adjust. [Just as they adjust twice a year to Daylight Savings Time in the US and elsewhere. Hopefully, when the Supreme Leader ends local time zones, he or she will also end DST.]
But after a few days, weeks, or at worst months, folks will have adapted and will learn that instead of setting their clocks to the local time while travelling, they just have to figure out when the local region sets their time-based events. "What time does the hotel serve breakfast?" for example. Airports and train terminals would publish signs that indicated what the local "Noon" time was, so people could track their days accordingly.
I imagine that this would result in either something like society had before time zones, where everyone set their "day" against the local noon, or people would set their "day" against the old time zones out of habit. After a generation or two, no one would really remember why their work day started at time X, just that "we've always done it this way," outside of Google searches and social studies books, at least.
What if we didn't use hh:mm:ss to measure time?
For that, I refer you to this list of alternate time measurement systems. It then links out to decimal time, which references French Revolutionary Time, and China time, among others. So there are historical precedents for not using hours, minutes, and seconds.
It would be a non-trivial task to convert to such a format, given how much of our world operates on the h:m:s principle. Phones, computers, clocks, etc. would all have to be updated or replaced...
Given how hard it has been to convince the US to adopt the metric system over the US Custom System, I suspect it would also be a Herculean task that would leave some nations on the new system and some still on the old, with many people confused as to how to convert between the two. (I know quite a few people that struggle with the simple task of converting between 12-hour and 24-hour times, much less a whole new system!)