MDN has it documented:

After postMessage() is called, the MessageEvent will be dispatched only after all pending execution contexts have finished. For example, if postMessage() is invoked in an event handler, that event handler will run to completion, as will any remaining handlers for that same event, before the MessageEvent is dispatched.

here's my test case (now that I've fixed it)

let workerSrc = `
let arr;
self.addEventListener('message', e => {
  switch (e.data.message) {
    case 'init': {
      console.log('worker: got init');
      arr = new Int32Array(e.data.buf);
      Atomics.notify(arr, 0);
      break;
    }
    case 'next': {
      console.log('worker: got next');
      arr[1] = 1;
      break;
    }
  }
});
`;
let blob = new Blob([workerSrc], {type: 'application/javascript'});
let worker = new Worker(URL.createObjectURL(blob));

(async () => {
  let buf = new SharedArrayBuffer(16);
  let arr = new Int32Array(buf);
  worker.postMessage({ message: 'init', buf });

  if (typeof Atomics.waitAsync === 'function') {
    await Atomics.waitAsync(arr, 0, 0).value;
  } else {
    // give the worker time to start
    await new Promise(res => setTimeout(res, 1000));
  }

  console.log('init');
  worker.postMessage({ message: 'next' });

  console.log('now we busy-wait, with arr[1] initially =', arr[1]);
  for (let i = 0; i < 1e8; ++i) {
    if (arr[1] === 1) {
      console.log('done! took until i =', i);
      return;
    }
  }
  console.log('never initialized');
})().catch(e => console.error('error', e));