I suppose that would technically be possible if the earth were flat and the sun revolved around it. But then, I think, the sun would still cast light on the east sooner than on the west. Or at least at a higher intensity.
Another option would be to make a spherical colony surrounding a star, like in that one episode of Star Trek TNG where they make up a reason to have Scotty still be alive.
As for the UK/South Africa issue, the sun doesn't rise and set vertically. That's why it's cold in Antarctica and un-cold at the equator. Behold!
If everyone just decided, "Ok, guys, when the sun is over Greenwich, it's noon everywhere," The meaning of the sun's position still wouldn't change. If the sun was high over England, and you called a friend in, say, China, it'd still be nightime there and they'd still be sleeping, despite the fact that it's technically noon. I guess everyone could change their schedules to compensate for this, but then one side of the world never gets to be awake during the daylight hours. Then they would inevitably complain, and the night side of the world would go to war with the day side for reclamation of lost sunshine, and the results would be CATASTROPHIC.
See what you fucking did? Your absent-minded wishful thinking has started World War 3, and I hope you're very, very proud of yourself.