Hi rscarlet,
you need to write an extension for that. This extension would add a button to the stage which toggles an internal variable (can be persisted through the engine) when clicked and if the variable is on, then a timer is set which moves the game to the next line. You probably want this to work for every time when a click is normally needed, not only for lines, right? Then you should listen to the event 
"wse.interpreter.next.after.nonext" on the game's bus.
In the 
index.html file you will find this code:
Code:
       <script>
            using("WSE.Game").run(function (Game) {
                
                console.log("Starting the game...");
                
                var game = new Game({
                    url: "game.xml",
                    host: typeof HOST === "undefined" ? false : HOST
                });
                
                game.start();
            });
        </script>
 
Use the 
game variable to add your button to the stage:
Code:
game.stage.appendChild(myButton);
 
Toggle a persistent variable when the button is clicked:
Code:
game.interpreter.globalVars.set("autoplayEnabled", !game.interpreter.globalVars.get("autoplayEnabled"));
 
Listen for the "nonext" event and set the timer:
Code:
var timerCode = null;
var autoplayWait = 2000;
game.bus.subscribe("wse.interpreter.next.after.nonext", function () {
    
    clearTimeout(timerCode);
    
    if (!game.interpreter.globalVars.get("autoplayEnabled")) {
        return;
    }
    
    timerCode = setTimeout(function () {
        game.interpreter.next();
    }, autoplayWait);
});
 
It might be a good idea to have a different autoplay wait time depending on the length of a line's content so that a user has enough time to read what's displayed. To do that you will have to differentiate between a "nonext" event triggered after a line command versus a "nonext" event triggered after another command.
You can do this by having a variable in your extension and listening to the line command's event:
Code:
var timerCode = null;
var autoplayMinWait = 1000;
var waitTimePerChar = 20;
var textLength = 0;
var isLine = false;
game.bus.subscribe("wse.interpreter.next.after.nonext", function () {
    
    var waitTime = autoplayMinWait;
    
    clearTimeout(timerCode);
    
    if (!game.interpreter.globalVars.get("autoplayEnabled")) {
        return;
    }
    
    if (isLine) {
        waitTime += waitTimePerChar * textLength;
    }
    
    timerCode = setTimeout(function () {
        game.interpreter.next();
    }, waitTime);
    
    isLine = false;
});
game.bus.subscribe("wse.interpreter.commands.line", function (data) {
    isLine = true;
    textLength = data.command.textContent.length;
})
 
Hope this helps!