Do I recall correctly that for the aborted 1977 'Phase II' series, there was a document (either in the series bible, or a memo that circulated?) where they nutted out these things and instructed the series' writers to use metric measurements? So that'd probably be the first time it was 'nailed down', so to speak.
Actually, no. The section on "Measurements" in the
Phase II bible is merely a slightly shorter version of the same section from the season-2 revision of the TOS bible, which states that they use metric (or light years/parsecs) for precise measurements and formal reports of information, but that more familiar units like miles can be used in vernacular, private conversation, or exclamations to make things clearer to the audience. So it was "nailed down" no later than April 1967.
It would also coincide with the US Government's own (ultimately failed) attempts to institute the metric system more generally in the mid 1970s. Although I'm led to understand that 'officially' the metric system is actually used in the States for a wide variety of things, including in the sciences.
The metric system has been the official measurement system of the United States since 1893. That's not a typo --
eighteen ninety-three. And ever since 1959, Imperial units have been legally defined based on metric units. The legal definition of a foot is 0.3048 meters. The legal definition of a pound is 453.59237 grams. Metric is the baseline.
Internationally, almost every country in the world uses the metric system (or rather, the International System of Units) as their primary, official measurement system; the US, Myanmar, and Liberia are the only countries in the world that don't mandate its use in official contexts, although the UK still makes some official use of Imperial measurements. Metric is the universal measurement system of science, engineering, medicine, global commerce, etc.