"Ain't" is short for "isn't", except that it isn't actually any shorter. Silly, ain't it?
In this statement there is an elision of "there", and of "who has", so it becomes: (There) ain't nobody (who's) got time for that.
Note the double negative, which here ain't a positive but is just for emphasis of the negation,
If you want to turn the double negative into a single negative, you can either remove the negative from "isn't" (as Copyright did), or (as I'd prefer) from "nobody", so for me it becomes "There isn't anybody who has (got) time for that." I like it better without "got".