Using High Resolution, I checked the automatic replay button but unchecked the timestamp replay checkbox. As soon as the replay file came out the timestamp is still there.
I believe this setting only applies to manually-saved replays, and for good reason: Without a timestamp on an automatically-saved replay, if you solve a level twice, the second replay will overwrite the first one. When you're saving replays manually, if you disable such an option, it's up to you to take care that this doesn't happen; but NL probably shouldn't automatically overwrite an existing replay under any circumstances.
Personally, I'm not sure why you wouldn't want the timestamps on auto-saved replays, but one solution that could allow the option to still work would be to append a number if the replay already exists, e.g. if replay.nxrb exists, save the file as replay_2.nxrb, then replay_3.nxrb, etc. I don't know how manually saving them works anymore due to using the auto-saving, but if it's automatically choosing the save location and filename instead of prompting the user with a system dialogue box, this might still be a good idea to prevent accidental replay overwriting.