It’s okay to adjust the timing. Missing the exact 12-hour spacing one time won’t stop nitrofurantoin from working. Antibiotics don’t suddenly stop working because the first two doses weren’t perfectly timed.
Your mom’s suggestion actually makes sense — taking your next dose tomorrow morning at around 9am, then 9pm, will put you back on a manageable schedule. What’s important is that after you reset the timing, you take the rest of the doses as close to 12 hours apart as possible.
So yes, the antibiotics will still work. It’s way better to get onto a schedule you can stick to than to force yourself to wake up at 2am for days. Just make sure you complete the full course.