Just tested it out, and the easiest way I can see (which seems to work in Chrome at least, but may need further testing) is setting a cookie.

On logout do something like setcookie('loggedout',1). You'll also need to do the opposite on login - unset($_COOKIE['loggedout'])

Then you just need some simple Javascript...

function readCookie(name) {

    var nameEQ = escape(name) + =;

    var ca = document.cookie.split(';');

    for (var i = 0; i < ca.length; i++) {

        var c = ca[i];

        while (c.charAt(0) === ' ') c = c.substring(1, c.length);

        if (c.indexOf(nameEQ) === 0) return unescape(c.substring(nameEQ.length, c.length));

    }

    return null;

}

window.setInterval(function() {

    if(readCookie('loggedout')==1) {

        window.location.assign('loggedout.html')

        //Or whatever else you want!

    }

},1000)

That'll check each second to see if the cookie is set. Magic.