Here's one version. I'm not a JavaScript developer, so there are probably easier ways to do this!

/*\
title: roundtime
type: application/javascript
module-type: macro

Round TiddlyWiki-formatted times to the nearest 15 minutes
\*/

(function(){
"use strict";

exports.name = "roundtime";
exports.params = [
  {name: "timestamp"}
];
exports.run = function(timestamp) {
    var hours = timestamp.substring(8, 10);
    var minutes = timestamp.substring(10, 12);

    if (minutes < 8) {
        minutes = 0;
    } else if (minutes < 23) {
        minutes = 15;
    } else if (minutes < 38) {
        minutes = 30;
    } else if (minutes < 53) {
        minutes = 45;
    } else {
        minutes = 0;
        hours++;
    }

    return timestamp.substring(0, 8) + hours + minutes + "00000";
};

})();

(If we wanted to dig into extracting seconds as well, we could round a bit more precisely. This is good enough for explanatory purposes!)