Although Colin Maclaurin didn't call it "Taylor's theorem," he presented it, attributed to Taylor, in his Treatise of Fluxions, 1742, see especially sections 751-754 where he introduced it. Maclaurin applied the Taylor series to a number of things, including using the series expanded around zero (now named after him) to characterise the maxima, minima, and points of inflexion of curves in terms of the signs of the relevant derivatives. For this discussion, see the Treatise of Fluxions, especially sections 857-860. Precisely what he says of the Taylor series' origin follows: "This theorem was given by Dr. TAYLOR, method. increm." (Treatise of Fluxions, section 751 = volume 2, p. 611)
Maclaurin may well have been aware of the Newtonian origins of this result, since he refers elsewhere to Newtonian work on series going back to the 1670s, and may thus simply be giving a convenient reference here rather than a statement of the origin of the result. Nevertheless, he does in effect call it a theorem of Taylor.
Judith V. Grabiner Flora Sanborn Pitzer Professor of Mathematics Pitzer College (909) 607-3160