Dominguez in History of the Convolution Operation poured through the original sources, and found many of Miller's and Gardner-Barnes's claims and citations to be inaccurate or erroneous. He devotes a separate section to main theorems associated with the convolution, where we read:
"On the other hand, another important theorem related to the CCO is the so-called convolution theorem. In general, the theorem establishes that the Laplace transform of the CCO (5) is the product of the Laplace transform of each input function. A similar result holds for (6) when the integral transform is the Fourier transform. In both cases, it is difficult to determine when both theorems appeared for the very first time. However, it can be said that the convolution theorem for (5) appeared in 1899 in Borel’s memoir about divergent series. For the case of (6), the convolution theorem appeared in the 1920 conference by Daniell about Stieltjes–Volterra products. In it, Daniell defined the convolution of any two measures over the real line, and then he applied the two-sided Laplace transform obtaining the corresponding convolution theorem. From it, he implied the result for the Fourier transform".
On the other hand, the rule of multiplying polynomials equivalent to convolving their coefficients (without the terminology or modern notation) appears already in the work of 12th century Islamic mathematician al-Samawal, see Katz, History of mathematics, 9.3.3.