Unlike French, there is no difference in underlying forms between oral and nasal vowels. It is widely believed that there is some nasalization of vowels before nasal consonants in English, so the obvious question to ask is, how can you possibly decide where if anywhere the nasalization is. The best approach is impractical but maybe worth a try. Get some hardware that measures nasal and oral airflow, and see when the nasal airflow starts to rise. Cohn did that in her UCLA dissertation, using a couple of California speakers, and compared those results to productions from French and Sundanese speakers, which have phonological nasalization. English is different, at least for her speakers: it is clear that there is a continuous increase in nasal airflow over the second half of a pre-nasal vowel, so the vowel isn't all-oral and it isn't all-nasal.
A cheap substitute for fancy machinery is micro-listening, using a bunch of recordings of word pairs like "made/mane", "pain/paid", "kin/kid", with the final consonant being either a voiced stop of a nasal. Using Praat or similar sound editor, trim the final consonant, and some contiguous portion of the preceding vowel. It is possible that the left portion of "pain" and "paid" become indistinguishable after editing out only a tiny bit of the vowel; or it may be that the two words are still recoverable even from just 40 msc of [pɛ] vs [pɛ̃]. If that is the case, you have evidence that the vowel is fully nasalized. With "mane / made" you also have the possibility of progressive nasalization, so an analogous question can be asked about "bane/bade" vs. "mane/made" – is there nasalization in "made" compared to "bade"?
In US English (some varieties) there is a marginal minimal pair, "can" and "can't", the latter having a nasal vowel. In UK English the vowels are totally different. The prospects for crossover to phonemic status are greater in US English, so barring an accent with surprisingly more nasalization, I expect that what they are telling you about "nasalization" is something about low-level phonetic details – which is best studied instrumentally and quantitatively. I think it is a misanalysis to say that vowels in English are "nasalized", rather they have some degree of coarticulation with an adjacent nasal consonant.However, this is just for your information: I'm not suggesting that you argue with your teachers.