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:

my_measure_unit

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