My answer to this question is to use thread micrometers for OD threads, combined with thread rings whenever possible, thread plug gages for ID threads.
Measuring threads is no simple task. Look at some of the contraptions that have been invented to measure them, and their expense.
A thread is one half of a simple machine. 3/4"-16 UNF-2A, for example, has a pitch diameter tolerance range from .7049 to .7094", for a total tolerance of .0045". Not that bad, until you consider that this is a measurement from gage line to gage line, half way down a tapered groove which is helical, perpendicular to the axis of measurement. Basically that means the pitch diameter is only one of the important aspects of a thread, and it isn't a straight-forward measurement.
Above, I recommended the use of thread micrometers. In my use, a thread micrometer will measure a thread pitch diameter +/- .0005". I would not trust it to accept or reject threads measured within .001" of min or max. Therefore, the 3/4-16 thread pitch diameter mentioned has to be maintained +/- .0013" in order to run parts without a thread gage, and I would get very uncomfortable making more than a few dozen parts like this even on a good machine.
This attitude is a compromise between doing it right, and doing it with what you've got. Doing it right entails meeting you contract obligations as specified by the purchase order and the engineering drawing, which either specifies or infers a particular thread standard like ISO 68. In actual practice, when you've got thousands of dollars on the line, a $250 set of thread gages which are calibrated by a good accredited lab which uses thread master gages to check and set your thread gages, is a necessary and practical choice.
So far I've only been talking about external threads. Measuring internal threads by any other means than hard gaging (thread plugs) is a tough choice to make. It's no problem to own and maintain thread plugs for common hardware sizes such as M8x1.25, #6-32, 3/4"-16, etc. When such a thread as M56x1.5 is specified, you're talking around $600 for a pair of thread plugs, with a couple of weeks lead time. If the quantity of these parts is low, you're really up against a wall.
In practice, good craftsmanship would tend to suggest that the machinist makes his own thread plugs on a lathe, for this job. Even a single thread plug gage can be a good asset here, sized to the mean of the mating thread's pitch diameter tolerance.
A machinist would make this thread plug and then monitor the product's thread using the feel of senses that come from experience making good threads. The machinist is not going to have good sense in making good threads unless he or she has spent a great deal of time in the company of genuine, calibrated hard gages and thread mics and thread wires.
Another good approach, in gageless situations of making such a thread as M56x1.5, is to obtain a mating part from the customer. There's no shame in asking for this, especially if you can explain to the customer that such a thread costs $600 minimum for hard gages, and that not everyone is as scrupulous in producing high quality threads as you are, and perhaps the mating external thread is close to tolerances itself.
You've seen this in your experience, because you are a bad-ass at measuring threads. And that is the moral of the story. Measuring threads isn't as straight-forward as you might think, and hitting tolerances requires a bit of study and a wealth of experience.