There are a few factors that explain why plea bargaining wasn't done until the 19th century or so in common law systems: trial procedure took very little time and resources, and there was little sentencing discretion. As a result, neither side was particularly incentivized to cut a deal. So a few changes needed to happen before plea bargaining became a rational option: trials needed to be costly enough to where prosecutors would rather not do a full blown trial, and sentencing flexibility needed to be able to offer defendants a promise of lesser punishment.

One of the earliest examples of plea bargaining as a prosecutorial case management tool came in 1801 when the English Parliament passed a statute that criminalized possession of forged bank notes. Previously, there was only the capital offense of forging bank notes, punishable by death. Once the lesser offense became an option, a strong incentive appeared for defendants accused of forgery to plea to just possession of forged bank notes. Similar lesser crimes were created in other contexts so that there was a lesser offense to plead guilty to.

At the same time, lawyers started taking over trials as trial procedure became more regimented. What was once just a presentation of an accuser and the defendant (without lawyers on either side) turned into a complicated and time consuming affair, with rules of evidence and procedure. Once that happened, prosecutors were incentivized to put together their solid evidence and just avoid the costs of trial.

By the end of the 19th century, plea bargaining had become routine for many crimes in both the United States and in England.

I picked most of this up from Langbein, Lerner, and Smith's History of the Common Law. I had it on my shelf at work, and figured I'd answer this question if nobody else beat me to it (criminal trials aren't really my thing).