Would it be normal to feel nauseous because you get a lot less sleep than usual over the course of a couple of days?

This would depend on many factors but yes, this is possible, especially if stress is the reason for the decreased sleep. The increased stress could also cause nausea. Hopefully, getting back on your usual sleep cycle will improve the nausea as well. If you need help to manage stress, see your doctor. Best of luck to you!

It would be unusual for lack of sleep to cause nausea. If the nausea persist or worsen, I would see your primary care doctor or a Gastroenterologist.