Voice actor changes, plus increased characterization
Firstly, the voice actors for battle droids changed between the movies and shows. One can also assume that there were technical differences between how they altered the voices between the different movies. Between these, there's enough reason to expect at least a noticeable change. But we're pretty good at finding similar sounding people, and these clearly seem intended to sound different, which means it was probably intentional.
Consider that in the first movie they didn't really serve a comedic role. It was mostly focused on their bad programming making them not-so-great soldiers. But they were otherwise not super chatty, didn't have much personality, and still a decent threat (to non-Jedi). As a result, it made sense to have them be fairly emotionless, robotic, unintelligent, and otherwise expressionless.
Now compare to the shows in particular, and you see how they wield a much more pronounced personality, are super talkative and silly, and downright have feelings and opinions on things. As a result, a sillier, less robotic, more lively voice was needed to convey them in the goofy/bumbling way they are now. They grew as characters into something more than they were before, and as a result became somewhat caricatures of themselves, parodying their traits from the first movie to an exaggerated level.
A related example of personality changes, compare R2-D2 of the original trilogy to that of the prequels and the shows. He goes from being a loyal droid to being something much more, and he gains a much deeper personality as a result.
If you want an in-universe reason, the comment by ThePopMachine is actually a pretty solid one. Their biggest shortcoming was their inability to understand/process/adapt to situations, because of their programming limitations. Realizing this, they may have decided to give them more intelligence and personality (though it's effectiveness is certainly in doubt).
EDIT: Thanks to my brother who recently rewatched the prequels for pointing out they didn't really serve any comedic role in Episode 1 at all. Just some occasional dry situational humor.