Yea, the short answer is that we never worry about Prime changing our test results with respect to cycling...
But! I actually can't remember about nitrite. I know that Prime (or any conditioner that uses the usual sodium thiosulfate) will not affect the ammonia reading because it just changes ammonia into ammonium and API and most other test kits will just see them as the same. Nitrate(NO3), as NOTG says, just stays the same regardless. But nitrite(NO2) does get neutralized (for 24 hours or so) by Prime and I can't remember if that makes a difference when reading a nitrite test kit result (I don't think it does but maybe somebody else will remember otherwise?)
Of course, there are other tricky things to be aware of: nitrite test results can usually not be trusted as much if there is also high nitrate in the water (I believe they are sometimes pulled to falsely higher results themselves by higher nitrate.) So what this means is that sometimes toward the end of cycles you think things are worse than they are but all of a sudden they get rapidly better!
~~waterdrop~~