There are a couple of answers to this question, but the short answer is religious and linguistic differences. I'm going to stick to the Christian bible because I think the reason that Jews don't use the Christian bible is fairly self-evident....

To answer your question about an authorizing body, I'm sure you know that there are several denominations of Christians, some of which, like the Roman Catholic Church, do have authorizing bodies that make determinations about the bible, and others, like various types of evangelical Protestants, where there is no centralized authority.

While all of the books of the New Testament were written within 100 years of the death of Jesus, they were not compiled together immediately as a "bible". Various attempts to compile the scriptures were made during the history of the Early Church, and ultimately Church councils agreed on the books of the New Testament in the late fourth century.

There has never been agreement among Christians over which books should be in the Old Testament. Different churches continue to disagree over the inclusion of some relatively minor Old Testament works, and each Church is subject to its own authority.

Other than disagreement, the second issue is language. The books of the Bible were written in Hebrew, Aramaic, and the Koine dialiect of Greek that was common in the Middle East during the Hellenistic and Roman periods. Obviously virtually nobody other than biblical scholars speak all of those languages.

The bible by necessity needs to be translated into languages that are actually spoken by Christians. Originally the bible was translated into Latin, but gradually there was a movement to translate the bible into vernacular (ie actually spoken) languages.

There were many partial translations of the bible into English in the Old English period, the Middle English period, and the Early Modern English period. These translations differ from eachother because English changed over time, because there is a tension between literal translations and conveying the accurate meaning, and different versions come down in different places on that spectrum.

Other than individual hierarchical churches, there's no authority that regulates bibles, and even within those churches there will need to be updates to reflect modern usages from time to time.