TL; DR. This is one of those cases where "first use" very much depends on what is meant. Depending on that, it can be ascribed to Babylonians (c. 1600 BC), al-Tusi (c. 1250), Briggs (1633), Newton (1669), Raphson (1690) or Simpson (1740).
The Babylonian rule (1800–1600 BC) for approximating square roots converted into modern notation gives the same formula as the "Newton's method", see Square Root Approximations in Old Babylonian Mathematics by Fowler and Robson. It does not come with even a vague anticipation of conceptions involved in the modern method. The same rule is described by Heron in Metrica (c. 50 AD), and is called "Heron's method" in older books. Heron also gives a related rule for cube roots, which does not reduce to the "Newton's method", suggesting that the connection is not too deep, see Remarks on Heron's cubic root iteration formula.
For the "warmer" part of history see Historical Development of the Newton-Raphson Method by Ypma:
"A method algebraically equivalent to Newton's method was known to the 12th century
algebraist Sharaf al-Din al-Tusi , and the 15th century Arabic mathematician Al-Kashi
used a form of it in solving $x^p- N = 0$ to find roots of $N$. In western Europe a similar method
was used by Henry Briggs in his Trigonometria Britannica, published in 1633, though Newton
appears to have been unaware of this."
Instead he worked from a more cumbersome perturbative method for solving polynomial equations laid out by Vieta in
De numerosa potestatum (c.1600), with additional inspiration from a very old method of false position, a.k.a. Regula Falsi. It is close to what is now called the secant method. A simplified version of Vieta's method was published by van Schooten in 1646 and reproduced by Oughtred in Clavis Mathematica (1647), this was Newton's source.
Detailed remarks on it are found in Newton's unpublished 1664 notebook, and by 1669 he linearized Vieta's successive polynomials to produce something convertible into the modern version in De analysi per aequationes numero terminorum infinita (better known for the first formulation of the method of fluxions). But, as with Babylonians, Heron or al-Tusi, there is little evidence that "his" method was the modern one conceptually. Here is Ypma:
"Newton's tract... is the first recorded discussion by Newton of what we may recognize as an instance of the
Newton-Raphson method (1.1), although the formulation differs considerably from the now
conventional form, the computations are much more tedious than in the current formulation,
and the method is given only in the context of solving a polynomial equation. No calculus
is used in the presentation, and references to fluxional derivatives first appear later in that
tract, suggesting that Newton regarded this as a purely algebraic procedure. In several other
instances Newton is known to have used more traditional methods and notations in an effort
to make his ideas more accessible to a wider audience, but there is no clear evidence that at
that time he perceived this particular technique as an application of the calculus or derived it
using the techniques of calculus."
The manuscript was not published until 1711, but private copies circulated earlier and the relevant content is reproduced in Wallis's Treatise of Algebra both Historical and Practical (1685). In Principia (1687) a similar procedure is applied to solving the Kepler's equation $x-e\sin x=M$, but again there is "no clear evidence that Newton associated his technique with the use of the calculus. There are numerous ways to derive this process that do not require the use
Raphson simplified the technique further in 1690 by eliminating successive polynomials completely, and making the scheme iterative. He already felt that the resulting method was different from Newton's. However, the modern conception with derivatives (fluxions) does not appear until Simpson's Essays on Several Curious and Useful Subjects (1740), where he does not credit any predecessors, and explicitly contrasts his calculus based procedure to the previous algebraic ones.