Hook in to the ClassicPress widget update process

apply-action-after-widget-update.js

jQuery(document).ajaxSuccess(function(e, xhr, settings) {
	
    // This is the data of interest.
    console.log(settings);
    
    // Identifier for your (or whatever) particular widget.
    var widget_id_base = 'text'; // text, search, my-widget, et al.

    // Check if updating a widget of this type.
    if((settings.data.search('action=save-widget') != -1) && (settings.data.search('id_base='+widget_id_base) != -1)) {
        // The data from the widget.
	console.log(settings.data);
	// Do amazing things here.
    }
    
});