If I have a conductor whose diameter is nonuniform, why is the current the same across every point of the conductor? I know that I=nqvA, where v is the drift speed, but doesn't that equation show that if my cross-sectional area increases or decreases, then my current will also change?
Your interpretation makes intuitive sense, because you're assuming that $v$, the drift velocity of the charge carries, is the same at all points along the wire. That assumption isn't right.
Water flowing through pipes (or a river) is a common analogy: A wide spot in the conductor is like a wide tank that the pipe runs into one end of and out the other end of. The water will flow more slowly inside the tank than the pipe precisely such that the volume of water passing through each section per second is the same.
If the flux of water volume (or the the electrical current $I$) were higher in one section than the section just before it, where would it be getting that extra water (those extra charge-carriers) from each second?
If the flux of water volume (or the the electrical current $I$) were lower in one section than the section just before it, where would the extra water (charge-carriers) from the prior section be going?