Reading Galton and Wright has indicated to me that even from the early days of considering correlation, there was some awareness that correlation is not synonymous with causation. However, who was the first (citable source) to say the familiar phrase "Correlation does not imply causation"?
tl;dr: A book reviewer with the initials F.A.D in a 1900 issue of Nature appears to be the first to publish the phrase "correlation does not imply causation."
Long form answer
Depending on whether you are looking for the exact words "correlation does not imply causation" (note also "correlation is not causation"), or just want the primary dive into the relationship between correlation and causation, a good answer is 18th Century Scottish philosopher David Hume in A Treatise on Human Nature, where he muses on the relation between correlation and causation. For example,
It is certain, that not only in philosophy, but even in common life, we may attain the knowledge of a particular cause merely by one experiment, provided it be made with judgment, and after a careful removal of all foreign and superfluous circumstances. Now as after one experiment of this kind, the mind, upon the appearance either of the cause or the effect, can draw an inference concerning the existence of its correlative; and as a habit can never be acquired merely by one instance; it may be thought, that belief cannot in this case be esteemed the effect of custom.
While Hume was nowhere near formalizing a causal calculus (e.g., Pearl's do operator), we can see in the above few sentences (abstracted from a whole section on correlation and causation), that he asserts that causal beliefs result from correlation between putative cause and effect of it, but only within a judicious model accounting for "superfluous circumstances" (backdoor confounding, selection bias, and differential measurement error, anyone?). Causal evidence demands correlation per Hume (both in the everyday sense, but also within the sciences), but neither naked correlation nor the absence of naked correlation is enough to establish a cause and effect relationship: in today's language we would say you need appropriate study design, and a causal calculus to deductively account for the structure of causal beliefs. Hume also made the notable contribution that causation can only be inferred, that what our senses give us is correlation.
As you indicate in your question, Sewall Wright is another worthy of consideration in answer to your question. One could make a claim that Wright originated the causal path analysis which led to both Pearl's work in formal counterfactual causal inference, but to other formal models of causal inference, such as Levins' loop analysis of complex causal systems in which every variable directly or indirectly causes every variable in the system at some future time. In his paper "Correlation and Causation," Wright noted:
One should not attempt to apply in general a causal interpretation to solutions by the direct methods.
Where "direct methods" are those measuring what I called "naked correlations" above set in the context of Wright's methodological contribution of path analysis (prefiguring the use of both directed acyclic graphs and signed directed graphs in causal inference). In other words: naked correlation is not causation.
If you are looking only for the exact phrase, Google ngram book search records the first appearance of that phrase in its corpus in a May 17th, 1900 review of racist eugenicist Karl Pearson's The Grammar of Science titled "BIOLOGY AS AN “EXACT” SCIENCE" in Nature by one F.A.D., who writes:
As the author [Pearson] himself elsewhere points out, correlation does not imply causation, though the converse is no doubt true enough.
Pearson himself does not use the phrase "correlation does not imply causation" but is grappling with the relation between the two in The Grammar of Science, for example:
All causation as we have defined it is correlation, but the converse is not necessarily true, i.e. where we find correlation we cannot always predict causation.
Coda: I fully agree with @BruceET's direct comment to your question. Without wanting to reify the WEIRD, I suspect that causal reasoning, and perceptions of correlation are pretty inherent to human cognition across societies and times (my cat informs me non-human cognition also).
D., F. A. (1900). Biology as an “Exact” Science. Nature, 62(1594), 49–50. [Collected in a May-October fassicle]
Levins, R. (1974). The Qualitative Analysis of Partially Specified Systems. Annals of the New York Academy of Sciences, 231, 123–138.
Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press.
Pearl, J. (2018). The Book of Why. Basic Books.
Pearson, K. (1900). The Grammar of Science (Second edition). Adam and Charles Black.
Wright, S. (1921). Correlation and Causation. Journal of Agricultural Research, 20(7), 557–585.