This is a supplement to the complete, technical answer by @guifa.
First, let me replace pronto with temprano. I checked the dictionary and found that pronto can be used to mean "early," but in the regional variant of Spanish that I know, it doesn't mean that at all -- it means "soon." Also, I'll adjust the word order to a more comfortable one. It feels awkward to me to have two adverbs right next to each other -- so I'll move one to a different part of the sentence. Finally, I'll use a singular sábado because that's the idiomatic way to talk about Saturdays in general in Spanish. The result:
El sábado no me levanto temprano nunca.
This would be equivalent to
Saturdays I don't get up early, ever.
You could think of no ... nunca as
... never ever.
Your teacher is right. If the nunca appears at the tail end of the sentence, it needs to have the basic negation word, no, near the beginning, because the nunca at the tail end is only there for emphasis -- rather like the "ever" in "never ever."
Now I'll make up a dialogue where the extra emphasis would make sense.
A: Ay, Mamá, ¿por qué sacaste cita con la doctora para el sábado a las 8?
Jeez, Mama, why did you make the doctor's appointment for Saturday at 8 am?
B: Así podremos ir al súpermercado a buena hora. Si no, habrá muchísima gente en el súper y se acaban las cosas, como el pan.
That way we'll be able to get to the grocery store at a good time. Otherwise, it will be really crowded and they'll be out of things, like bread.
A: Pero ya sabes, el sábado no me levanto temprano nunca. El sábado es mi único día de descanso.
But you know that I never get up early on Saturday, *ever*. Saturday is my one day off work.