The author uses the term "uniformly regular" and I get the idea of it's meaning through the context, yet the phrase is used as if could also have a precise mathematical meaning.
Is there a definition of uniformly regular signals?
The first chapter and the preface are available at the books website
This is indeed not precisely formulated in the book. Anyway, here is what I can make of it: Regularity usually refers to the degree of differentiability. What degree it refers to is not of central interest to the author I think. The purpose of the formulation seems to be to give the reader an intuition of the meaning of decay of the Fourier Transform (high order of differentiability <-> fast decay of the Fourier coefficients).
I am quite sure there is no precise definition of a uniformly regular signal. The mathematical equivalent should be continuously differentiable of a certain degree.