Technically the US measurement system is metric since the Mendenhall Order of 1893 reestablished all customary units as conversion factors of metric units. In 1933 the ASA redefined the inch to be exactly 25.4mm, following the lead of the British Standards Institution in 1930 (precision was increasingly important for manufacturing, and the previous value of 25.40005mm had become impractical). The international yard and pound were officially adopted by the US National Bereau of Standards (now NIST) in 1959, the Metric Conversion Act was passed in 1975, and finally EO 12770 (1991) required all agencies of the executive branch to transition to metric units.
So, from one point of view we’ve been transitioning to metric since 1893 and it’s still not done. From another, the inch is just a metric unit as its length is officially defined in millimeters (all customary units are now based on SI units), therefore the conversion is complete.