Some progress! 

In ‘a Rudimentary Treatise on Masonry and Stonecutting’ published in 1849, the author Edward Dobson writes - 

 ‘The curved rulers manufactured in Paris of thin veneer, and sold under the name of French curves, are very useful for drawing in between points previously determined small portions of elliptical or other curves, which cannot conveniently be struck from centres.’  (ie they had no single/common radii - or indeed any radii except at a single point.  You can tell I'm not a mathematician.....)

As I have been unable to find an earlier reference (so far), and this is a book from an English author and publisher, it seems fair to conclude that these templates were so-named because of their country of origin. 

David Browne

I've also been searching fairly fruitlessly for French Curve information as part of a Taxonomy of Automotive Design-related topics which I'm writing, but less about their history and construction (which I've summarized thus: Although the first use of the name ‘French’ seems uncertain, the defining of these curves can be traced back – via numerous C.19th French mathematicians - to Pascal, Descartes and Pierre de Fermat (most popularly famous for his ‘Last Theorem’) who in the C.17th devised an original method of finding the greatest and the smallest ordinates of curved lines which is still acknowledged as the foundation for the modern theory of such curves.)  Someone more interested in that aspect may like to delve more deeply.

What I've failed to find are any answers to the apparently 'simple' questions: when, why, where, who etc first called the curves designed by Burmester 'French'.  

A few years ago, there was a wonderful exhibition of the streamlined French cars of the 30's called 'French Curves' but I haven't found any evidence that such templates were used to create their extraordinary forms in the design process.  More likely a combination of splines and a creative and confident eye.

I'd welcome any answers, leads or clues!

David Browne