As long as you use any cache plugins, the plugins will cache everything, including the Measurement cookies and the Measurement switch will show your chosen Measurement only after cache is deleted from plugin.
Exclude this cookie from being cached for the widget to work:
NOTE: You can set to exclude all these cookies from your caching system (applies to multi currency, multi-measurement widgets)
my_custom_curr my_custom_curr_pos my_custom_curr_symbol my_measure_unit
EX: How to exclude from Fastest Cache plugin
Ex: How to exclude from WP Rocket plugin