These are only guesses, I don’t know anything about the plugin that you are referring to.
My first guess at why fields are not saved in those widgets is because they are not saving the widgets in a standard fashion. With widgets ACF save fields on the widget_update_callback hook. If for some reason this hook is not being fired by WP then ACF would not update the values.
My second guess, if this hook is being fired, is that ACF can’t figure out the correct ID to save the values to. The 4th parameter of the callback is $widget and ACF saves the fields to acf_save_post( "widget_{$widget->id}" );
then again, that values might be getting saved, and ACF can’t find them again because of some other issue that is making it hard for ACF to get the values that were saved when it reloads. I can’t be sure.
I’m not the developer, but ACF is designed to work with the standard way that WP does things. If the widget plugin does things in some other way then ACF may never work with it.
