Keep in mind, that movie came out in 1994 or 1995. Caller ID was brand new and only available in certain areas at the time.
My guess is that Simon used a computer hacker to hack into a few dozen of phone companies across the globe. He instructed his computer hacker to make a number in Sidney divert to a number in Moscow. Then the number in Moscow divert to Egypt. And so on. The entire time the hacker was setting all this up, intense music was playing and he was spouting a lot of humorous catch phrases.
I'm not sure what kind of system he used to pull all this off, but I'm pretty sure it was very graphical and had various maps of the world with colorful lines pointing to and from all the locations that he was diverting. As the police initiated the trace, their computers displayed a similar map with the colorful lines slowly zooming around to the various diversion sites.