A WordPress plugin conflict, rebuilt so you can see it. One plugin update loads its own old copy of jQuery over WordPress's, and every script that ran after it breaks: no error on the page, the button just does nothing. Switch between the broken site and the fixed one and press the button in both.
Pick a time, then press Book.
WP_DEBUG_LOG and open the browser console on the broken page. Here the console names it: Uncaught TypeError: $(...).datepicker is not a function, and the network tab shows jquery-1.12.4.min.js loaded by the updated "Popup Pro" plugin after WordPress's own jQuery 3.7.add_action('wp_enqueue_scripts', function(){ wp_dequeue_script('popup-pro-jquery'); wp_deregister_script('popup-pro-jquery'); }, 100);