There are two ways this could go. Most of the answers assume that all timezones but one (generally GMT/UTC) are abolished and all our clocks are set to the same time. People will just have to get used to having breakfast at midnight, or going to bed at 2pm, if they are unlucky to live in a place where that is the custom.
But the other way would be to return to astronomically correct time, wherever you are. In normal rural life, ... no problem. If you have to communicate with another location, you need to calculate the correct time wherever they are - for example, in Bristol you'd better turn the TV on at 5:40pm to catch the 6 o'clock News from London.
Then there's the discrepancy between solar time and mean time, defined by the Equation of Time. Your clocks (which drift) run on Mean Time, but your sundial (which is accurate) reports solar time, and they can be up to 15 minutes apart in either direction at different times of the year, so even in one location you need to convert from one to the other with a calendar and a pocket calculator...
What day it is - now that's a whole different problem :-)