In my survey, I use the first question as a button to show an additional piece of information
Great, this gives you a trigger to measure the first timestamp. You can store it in JavaScript (temporary in the browser window) like this:
var ts0 = Date.now();
I made it automatic to jump to the next page when the second question is answered.
Then use the JavaScript-check for missing answers as second trigger. And also store the time difference in that function.
function storeTime() {
    var ts1 = Date.now();
    // Write into internal variable (that must have been dragged on the page)
    document.getElementById('IV01_01').value = ts1 - ts0;
    return true;
}
SoSciTools.questionnaire.attachCheck(storeTime);