Voldemort does not have a nose because his physical appearance changed due to him delving deeper and deeper into the dark magic. That made his face more serpentine and his nose changed into two snakelike slits. Some theories say he lost his body, experimenting with dark magic. Hence, he created his own body.
The most likely reason why Lord Voldemort’s nose disappeared and he devolved into such an evil-looking creature is that as he dabbled deeper into the Dark Arts, his appearance gradually became as distorted as what remained of his twisted soul.