This is a script for Drafts.
The timestamp format is simply ‘hh:mm’ and is inserted right at the cursor location.
let loc = editor.getSelectedRange()[0],
now = new Date().toTimeString().slice(0, 5);
editor.setTextInRange(loc, 0, now);
editor.setSelectedRange(loc + 5, 0);
editor.setSelectedText('');
// activate is not required if the action is launched via a keyboard button
editor.activate();
Published in the Drafts directory.