As an uncontested divorce lawyer in Illinois, I have handled hundreds of uncontested divorces in many counties in Illinois, including Cook, Champaign, DuPage, Grundy, Kane, Kendall, Lake, McHenry, Sangamon and Will. Out of the hundreds of uncontested divorces I've handled, literally at least 99% of them have required only one court date.
However, there is a very small chance that your uncontested divorce could require more than one court date. Some counties require at least two court dates, the first one just being for a lawyer to get a prove up date (with the prove up being the second court date). Additionally, people with very unique circumstances sometimes have settlement agreements with which a judge is uncomfortable, and the judge asks people to come back to court again after making some adjustments (but again, that is VERY rare).
When I handled a flat fee uncontested divorce, I can advise my clients if they want to do something that could cause the divorce to require more than one court date.