It was definitely “monkey” when I first heard it over fifty years ago but it was always as an answer to “what time is it?”
Never heard “Mickey”, it’s always “monkey”.
I’ve also only heard “monkey”.
The Historical Dictionary of American Slang gives “monkey’s ass” as a synonym for “monkey’s uncle”, in the phrase “I’ll be a monkey’s ass”. But I suspect that’s unrelated, and “half past a monkey’s ass” was just a bit of wordplay based on the rhyme and rhythm of the phrase.
The first Mickey Mouse watch was made in 1933, so if the phrase predates that date, it would definitively refute the “mickey” version.
ETA Although it’s certainly possible, even likely, that someone somewhere modified the “monkey” phrase to “Mickey” with reference to Mickey Mouse watches.