It won't delay the interview date, but it might increase the time between the interview date and oath date by 1 or 2 months, because only judicial oaths can include the name change, and judicial oath ceremonies are held less frequently than regular administrative oaths.
However, if in your location all oaths are judicial oaths, there would be no difference.
Note that if the name change is done in court independently of the citizenship process, that in itself can take a couple of months, in addition to costing hundreds of dollars.