Speed of a background check depends entirely on the agency doing it and how they prioritize it along with their other responsibilities. I do background checks for a variety of organizations. I have done them in as fast as a few hours if the case is not particularly in depth. The longest it's ever taken me to do just a standard background check on anyone is 6 weeks. If there's anything in your background that would require a more in depth look, that requires me to access more records or call people or physically go somewhere to obtain them, and that would obviously make it longer.
If you've never changed your name before, don't have a criminal record, aren't delinquent on any debts, etc. a good turnaround would be 1-2 weeks. If the agency doing the background check is slammed, 4-6 weeks. Generally. If they've just got one dude doing them on a first-come first-served basis, YMMV.
If you have contact information for the place doing the background check, I would call and ask what the typical turnaround is. Then I would call the court directly and see what they recommend. If you live in a state where you're going to have to go before a judge, I would try to make the date as early as possible since they're usually booked out a few weeks to months in advance.