Get Last Modified Date from a Public Accessible File (Async). No external libraries required. ECMAScript 5.

get_file_last_modified_date.js

/*
 *
 * Fiddle: https://jsfiddle.net/w79L4wan/10/
 */

function ready(callback) {
  // in case the document is already rendered
  if (document.readyState!='loading') callback();
  // modern browsers
  else if (document.addEventListener) document.addEventListener('DOMContentLoaded', callback);
  // IE <= 8
  else document.attachEvent('onreadystatechange', function(){
      if (document.readyState=='complete') callback();
  });
}

ready(function (){
  updateFileDate();
});

function fetchHeader(url) {
  var xhr = new XMLHttpRequest();
  xhr.open("GET", url, true);
  xhr.onload = function (e) {
    if (xhr.readyState === 4) {
      if (xhr.status === 200) {
        var lastModified = xhr.getResponseHeader('Last-Modified');
        formatAndUpdateDate(lastModified); 
        return;        
      } else {
        console.error(xhr.statusText);
      }
    }
  };
  xhr.onerror = function (e) {
    console.error(xhr.statusText);
  };
  xhr.send(null);
}

function formatAndUpdateDate(dateString) {
  var parsedTime = new Date(dateString).getTime();
  var a = new Date(parsedTime);
  // If you prefer non-numeric month:
  // var months = ['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov','Dec'];
  // var month = months[a.getMonth()];
  var year = a.getFullYear();
  var month = a.getMonth() < 9 ? '0' + Number(a.getMonth() + 1) : Number(a.getMonth() + 1);
  var date = a.getDate() < 10 ? '0' + a.getDate() : a.getDate();
  var hour = a.getHours();
  var min = a.getMinutes() < 10 ? '0' + a.getMinutes() : a.getMinutes(); 
  var sec = a.getSeconds() < 10 ? '0' + a.getSeconds() : a.getSeconds();

  var ampm = hour >= 12 ? 'pm' : 'am';
  hour = hour % 12;
  hour = hour ? hour : 12; // the hour '0' should be '12'

  var formattedDate = month + '/' + date + '/' + year + ' at ' + hour + ':' + min + ampm;

  var el = document.getElementsByClassName("last-updated-date")[0];

if (el !== undefined) {
    el.textContent=formattedDate;
  } 
}

function updateFileDate() {
  var url = document.getElementById('file').href;

  if (url !== undefined) {
    fetchHeader(url);
  }
}