ACF + WPML: How to translate date fields?

You can use the global $wp_locale to get the month translated in each language. You just need to get the month number (01 to 12) from the Acf field.

You have the functions in the WP_Locale class get_month() and get_month_abbrev()

global $wp_locale;
$month = $wp_locale->get_month(04); // Output april in english
$month_abbrev = $wp_locale->get_month_abbrev($month); // Output Apr in english