At what point did the 60 seconds = 1 minute and 60 minutes = 1 hour system become the system used by everybody. Were there other timing systems used by people before they came into contact with the current system?
That system has been in use since since the times of the Babylonian empire, who used the duodecimal system, which became the basis of the timekeping system. It was adopted by all civilizations around the area, including the Roman empire, then the Christian and Muslim civilizations. And with European colonization of the Americas and Oceania it spread also there, so eventually oly some Asian countries didn't use it - for example Ming China used to divide day into 10 parts.
However, while the basic system remained the same, these were important differences - in the Roman empire day was divided into 12 hours, and because the lenght of the day depends on season, the length of the hours also varied. Seconds didn't exist in the ancient times, because they didn't have the devices to measure them, watches thatwre precise enough were invented only in modern Europe. Once a second also used to be divided into 60 terces (pars minuta tertia), but this is no longer used.