I came across this question while trying to figure out when the "law of propagation of error" was first stated, which resulted in this question: When was the "Law of Propagation of Error" first stated? which now contains a reference to your question.
Your question is about when error propagation become prominent (rather than about when it first appeared). That part I can answer by saying that Raymond Birge in 1939 said that the question of how to assign an uncertainty has been discussed for decades but the subject matter of error propagation is one for which "many scientists still fail to avail themselves" and that "others frequently use the theory [of error propagation] incorrectly and thus arrive at quite misleading conclusions".
Raymond Birge was a prominent enough physicist, as he was the head of Berkeley's Physics Department and was regularly in contact with people like Gilbert Lewis, Robert Oppenheimer, and various other famous Nobel laureates or future Nobel Laureates. Therefore if there was people widely doing error propagation in a way that was "correct" in his eyes, he would have known about it or had access to the people who could advise him of the truth on the matter. In fact it was his frustration with inconsistencies in how people made measurements and reported fundamental physics constants that lead him to publish a series of papers about error propagation. People making measurements of fundamental physics constants would have been elite physicists (having enough money to do such experiments, and having also enough education to do such experiments), so if they were doing error propagation wrong, you can imagine that they were also teaching it wrong to the next generation of physics student.
As my above linked question describes, the chronologically next paper which comes up most often is this 1966 paper which says in the first sentence of the abstract:
The "law of propagation of error" is a tool that physical scientists
have conveniently and frequently used in their work for many years
So by 1966, error propagation seems to have been quite popular. Perhaps Birge's papers in the 1920s and 1930s were what made error propagation popular. I already mentioned that Birge was the head of Physics at Berkeley and therefore taught many students that became prominent future scientists and had influence on a lot of other people too, but if that argument is not convincing enough you can see that between 1930 and 1940 his paper "Probable values of the general physical constants" was cited in Physical Review or Reviews in Modern Physics papers by I.I. Rabi (from Rabi Oscillations), Marie Curie (who needs not further words), Harold Urey (from the Miller-Urey experiment), Irving Langmuir (the namesake of the ACS journal Langmuir), Robert Millikan (from the Millikan oil drop experiment), E.E. Witmer (from the Wigner-Witmer rules), Walther Gerlach (from the Stern-Gerlach experiment), and about 70 other papers between 1930 and 1940 alone: this I think is enough to say that Birge was influential in science and probably were his papers about error propagation in the 1920s and 1930s.
Birge himself said in the 1939 paper that the idea of error propagation was not new, and he frequently cited a 1894 book by Mansfield Merriman called "Method of least Squares" for which a 1910 version can be found here. So perhaps Merriman (a civil engineer) popularized error propagation enough for it to be noticed by Birge (but insufficiently enough for Birge to be complaining about most people either not being aware of the theory or using it wrong), and then Birge popularized it to many of the prominent or to-become-prominent scientists of his time.
Conclusion: The mathematics of error propagation was discussed earlier by people like Mansfield Merriman, but even in 1939 it was not well-known by many people and was used incorrectly by many people, but Raymond Birge published several papers in the 1920s and 1930s on the topic of uncertainties in measurements, which were cited by many of the top scientists of the time. By 1966 error propagation was considered to be quite common.