I got hardwood installed and just realize that the stair nosings are not exactly flush with the engineered hardwood on the treads. What is the tolerance and how can I test it?

its manufacture specific, each has their own. How bad is it? That's one reason i prefer solid treads over the flooring/nose cap way.

the difference is about the thickness of a cardboard from a cereal box.

The milling can easily be off that much from a nosing to the flooring, even the flooring edge to flooring edge with a square edged prefinished engineered.

Thanks for the answer.

99% of moldings are farmed out to a third party. thats why they cost an arm and leg, rarely are perfect color match and are not dead on milling to the floor.