In some areas all or nearly all the oaths are judicial oaths, so there won't be any delay. In other areas there may be a slight delay for a judicial oath, but we're talking about maybe a few weeks of delay, not several months.
However, after thinking about it I must say that adding a middle name is a really good idea to get a name change without too much disruption. The documents you have like driver's license, degrees and diplomas, etc. would already have your first and last name so there is no immediate problem with them. And if you added the name to have a shorter/more Americanized name, you could have people in daily life like work and school call you by your middle name and you can put the middle name on your resume when applying for jobs, while your family still calls you by your same first name.