Instead of working form the theorical diameter of the wire when making the coil, I am interested in measuring it as it unwinds from it's original spool (more accurate).
I'm skeptical that your measurement accuracy would be better than the manufacturing accuracy. Have you measured the actual thickness, and does it vary (between spools, or within a spool) by enough to matter?
If your goal is just to wind wire onto a coil former then I would have thought you effort would be better spent designing the winding mechanism so that the wire stacked itself correctly. Doesn't a conventional winder sort this out for you? I'm surprised that it's necessary to do anything with any great precision when feeding the wire on, or that there are any benefits in knowing the wire thickness precisely.
If you really do need to know the wire thickness as an academic exercise then I'd use a micrometer. If it's necessary to measure this repeatedly at different places along the wire then I'd pass it between a pair of fixed/floating rollers and use a micrometer to measure the position of the floating roller. But this still feels as if you're solving the wrong problem.